Wikiversity:Colloquium

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== Paid part-time Administrative Officer position with WikiJournal ==

[[WikiJournal User Group|WikiJournal]] is an open-access, nonprofit scholarly publisher hosted on Wikiversity. It uses the wiki format to allow authors to write and develop scholarly articles directly online. Submitted articles then undergo formal academic peer review and, when accepted, are published as citable journal articles while remaining freely accessible.

WikiJournal is seeking expressions of interest in a new part-time '''Administrative Officer''' position. The officer would support the project's day-to-day organizational work, including:

* Setting up and administering a system for tracking contractor hours and issuing payments.
* Coordinating the contracting, onboarding, timekeeping and payment of technical editors and other contractors.
* Maintaining financial, contractor and administrative records.
* Assisting with grant administration, budget tracking and reporting.
* Following up on board action items, deadlines and recurring obligations.
* Supporting recruitment and other operational tasks as needed.

This is a remote, flexibly scheduled contractor role averaging approximately '''6 hours per week'''. Compensation is proposed at '''US$25 per hour''', with up to approximately '''320 hours (US$8,000) available over the year'''.

Applicants may be based internationally, although appointment will depend on WikiJournal being able to establish a practical and legally appropriate contracting and payment arrangement in the applicant's country of residence.

Relevant experience may include administrative organization, bookkeeping or financial administration, contractor or personnel coordination, nonprofit or grant administration, Wikimedia participation, academic publishing, and open-access work. Applicants are not expected to have experience in every area.

For the complete role description and further details, see [[WikiJournal User Group/Administrative officer]].

Those interested should reply in the '''[[Talk:WikiJournal User Group#Call for interest: Administrative Officer|Call for interest: Administrative Officer]]''' discussion by '''30 September 2026''', briefly describing their relevant experience, any connection to WikiJournal or related communities, and their general availability. Expressions of interest received later may still be considered if the position has not been filled. Recommendations of potentially suitable candidates are also welcome. 
[[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:00, 10 September 2026 (UTC)

== Server switch - Your wiki will be read-only for a short time soon ==
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[[m:user:Trizek (WMF)|Trizek (WMF)]] 12:58, 15 September 2026 (UTC)
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== Wikidata Sitelink changes in Recent Changes and Watchlist ==
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Editors on group 1 wikis, including Wikiquote, Wikibooks, Wikiversity, Wikivoyage, and Wikisource, will no longer see notifications on [[Special:RecentChanges|Recent Changes]] and [[Special:Watchlist|Watchlist]] when a link to ''another'' language version of a page is added, removed, or changed in its Wikidata item. This change will begin on '''October 5 at 07:00 UTC''' as a two-week trial to reduce high-volume, low-value notifications and make higher-value changes easier to identify. Editors who rely on these notifications should be aware of the change and can share feedback or concerns on the [[m:Wikidata For Wikimedia Projects/Clearer Wikidata Edit Summaries/Hide Language Link edits|project]] or [[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|talk page]].&lt;br/&gt;Please reach out if you have any questions, thank you! -- [[User:Danny_Benjafield_(WMDE)|Danny Benjafield (WMDE)]] ([[User_talk:Danny_Benjafield_(WMDE)|talk]])  sent via [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:42, 1 October 2026 (UTC)
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== Demote [[Wikiversity:CheckUser policy]]? ==
I am proposing that we demote that policy to a standard page, mostly for two reasons:
# Considering the few amount of custodians and the size of this wiki, I'd say that this project is considered too small to have any CUs, and relying on the stewards would be a more logical choice.
# We haven't elected two CUs in a few years.
[[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 20:53, 1 October 2026 (UTC)

:Reasonable. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:09, 4 October 2026 (UTC)
:I don't see much point in demoting it. Even if we are not currently (or recently) using the policy, it is there should a need ever arise. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 18:24, 23 October 2026 (UTC)

== General Questions and Reference Questions  ==
&lt;p&gt;Hey everyone,&lt;/p&gt;
&lt;p&gt; My apologies in advance if these questions are oblivious or their responses readily apparent, but I figured I'd ask just to be safe.&lt;/p&gt;

&lt;p&gt; '''Questions'''&lt;/p&gt;
&lt;p&gt; 1. Should pages have inline citations and a reference section? A [[Fundamentals of Neuroscience/What is Neuroscience|majority of pages]] I have explored have neither, while [[Neurobiology of emotion|others have a more traditional approach to references]].&lt;/br&gt;
2. Are there widely-used templates for weasel-words, references, etc.? 

&lt;/p&gt;
'''CONTEXT''': I migrated from Wikipedia, so I'm not entirely sure what the norms/roc are here.&lt;/br&gt;

Thanks in advance,&lt;/br&gt;
[[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 18:40, 12 October 2026 (UTC)

:You're asking fine questions. I posted {{tl|welcome}} to your talk page which I think will help you. To answer your direct questions:
:#This project spun out of [[:b:]] and both here and there, you can post original research. This does not mean that you can post just anything, so if you want to post an educational resource and you want to justify claims you make with inline citations, that's great. If you want to include a bibliography, that's also quite helpful for readers. Generally, the way that Wikiversity works in practice is that individual users make a resource and then tend to work on it solo and have broad latitude about what you can do and how it looks.
:#You can take al look at [[:Category:Templates]] and the subcat [[:Category:Maintenance templates]]. At Wikipedia, it's very common to point out issues with tone or uncited claims. That's not really as common here. We [[:d:Q7780480|do not have an equivalent]] of [[:w:en:Template:weasel]], but we do have a [[Template:Citation needed|local version]] of [[:w:en:Template:Citation needed]].
:―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:31, 12 October 2026 (UTC)
::Thanks for the heads up on both accounts; I'm happy to join forces with all y'all!
::[[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 23:39, 12 October 2026 (UTC)

==Agent editors==
What about inviting agents to edit this wiki? And they would need a human being responsible for them. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 06:31, 17 October 2026 (UTC)

== October 2026 Wikimedia Café: legal issues on Commons, and WMF Board selection process ==
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Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in October. In a change from the usual Café practice, this month's sessions will have different topics. Anyone who meets the Café attendance criteria is welcome to attend either or both sessions.

Session A will happen on Saturday, 24 October 2026 at 15:00 UTC ([https://zonestamp.toolforge.org/1792854000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe. This session will feature [https://meta.wikimedia.org/wiki/User:PBradley-WMF Wikimedia Foundation lawyer Phil Bradley-Schmieg]. We will discuss legal issues on Wikimedia Commons, including complexities of international law. (I am hoping that in a future month, a similar Café session will be offered at a time friendly to Asia and the Pacific.)

Session B will happen on Sunday, 25 October 2026 at 03:00 UTC ([https://zonestamp.toolforge.org/1792897200 timestamp converter]), at a time friendly to Asia and the Pacific, we will discuss [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposed changes to how Wikimedia Foundation Board candidates are selected]. The latest proposals are much different from the earlier July 2026 proposals. The due date for feedback on current proposals is 27 October, and Wikimedians are welcome to participate on the [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposal talk page] regardless of Café attendance.

Please see the [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Café] page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!

To subscribe or unsubscribe for Café notices on your talk page please [https://meta.wikimedia.org/wiki/Global_message_delivery/Targets/Wikimedia_Caf%C3%A9 go here].
&lt;br /&gt;
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&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;༄ᨒ𖠰 Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 06:22, 18 October 2026 (UTC)

== A new cosmological model ==
I would like to publish independent research that challenges the standard cosmological model. Does this fall within the scope of the &quot;Original research&quot; space? And what precautions should I take to ensure the page is not considered an inadmissible personal work, and is not open to modification by other users? [[User:Wb525|Wb525]] ([[User talk:Wb525|discuss]] • [[Special:Contributions/Wb525|contribs]]) 11:53, 20 October 2026 (UTC)
: This sounds like it could be [[Wikiversity:Original research|original research]]. Some things to note:
:* Sometimes what an author considers original research turns out to be original theory
:* Some qualities that may help to ensure viability: objectivity, use of simple language, careful use of peer-reviewed citations, and minimal use of genAI with careful attribution/acknowledgement if AI is used per [[Wikiversity:Artificial intelligence]]
:* Consider starting with a sub-page in user space e.g., [[User:Wb525/Subpage name]]
:* As an open knowledge project, Wikiversity content is openly editable by others
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:05, 20 October 2026 (UTC)

== Interface administrators policy==

This is a discussion of potential changes to [[Wikiversity:Interface administrators]] policy.

Please refer to: &quot;For now, Wikiversity does not have a need for permanent or long term interface administrators. Accordingly, interface administratorship may only be granted temporarily by bureaucrats (between 14 and 120 days based on discussion, at the discretion of that bureaucrat and perceived need).&quot; --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:14, 23 October 2026 (UTC)

=== Temporary rights duration ===
I propose that temporary rights be granted for a duration of from one day to one year. (to conform with selections in [[Special:UserRights]] in the pull down menus.) --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:14, 23 October 2026 (UTC)
==== Discussion ====
*{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:56, 23 October 2026 (UTC)
* {{support}}. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC)
* {{support}} as proposer.--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 16:30, 23 October 2026 (UTC)
* {{neutral}}; if we're changing the time period again, this time to align with the drop-down options, then I suggest that the policy require that nominators request one of the following time-frames supported by a corresponding plan of work:
** 1 day
** 1 week
** 1 month
** 3 months
** 6 months
** 1 year
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:01, 25 October 2026 (UTC)
::Yes, that was my intent. I agree that a nom should request a duration with a &quot;corresponding plan of work.&quot; [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:17, 28 October 2026 (UTC)
* {{support}} sensible proposal. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 15:20, 29 October 2026 (UTC)

=== Temporary rights extention ===
I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:14, 23 October 2026 (UTC)
==== Discussion ====
*{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:56, 23 October 2026 (UTC)
* {{support}} for more than one extension. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC)
* {{support}} as proposer. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 16:30, 23 October 2026 (UTC)
* {{Neutral}} because I'm not sure what the practical difference is between applying for an extension and applying for the right. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:04, 25 October 2026 (UTC)
*: On a nomination or IA request, a member may question what timeframe a candidate may request, or the candidate may state what timeframe they need for IA rights. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 02:51, 25 October 2026 (UTC)

=== Permanent IA ===
There has been a suggestion of permanent IA rights. Another suggestion is that there should be at least two members with such rights. Please express your preference. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:14, 23 October 2026 (UTC)
==== Discussion ====
*{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:56, 23 October 2026 (UTC)
* {{support}} for having at least two members because there shouldn't be only one. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 13:59, 23 October 2026 (UTC)
* Conditional {{support}} with the understanding that there be a minimum of 2 (if only 1 is remaining, the other perm is set to temp until a 2nd becomes active) and perm can only be granted after having a temp. I also think a perm nom should have a requirement to demonstrate need - for example show activity logs so that the community can assess that IA is needed long term. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:55, 23 October 2026 (UTC)
*: You mean that we may have at least 2 permanent IAs, but if one is removed, the other one's is set to temporary until another IA is appointed, makes sense? [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 20:34, 23 October 2026 (UTC)
*::Yes, I meant that if the second IA is no longer active, the remaining IA would be reset to 1 year temporary. That should give enough time to find a second perm IA. That 1 year could be extended, if it takes more time to find a second. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:15, 24 October 2026 (UTC)
*:::Once the community approves a perm IA, there is no need to renominate. A crat can just reset the time to temp until the conditions are met, and reset to perm when we have two or more. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:25, 24 October 2026 (UTC)
* {{Neutral}}. Perhaps if there are users who make effective use of IA rights across multiple temporary periods, permanent rights could be useful. My hunch is that is unlikely in the foreseeable future for en.wv to manifest two users who simultaneously maintain that level of engagement. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 25 October 2026 (UTC)

== Server switch - Your wiki will be read-only for a short time soon ==
&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2026-10-28|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-10-28T15:00|en}} {{#time:H:i e|2026-10-28T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 
You can contribute to the [https://meta.wikimedia.org/w/index.php?title=Special%3ATranslate&amp;group=Centralnotice-tgroup-read_only_banner&amp;task=view&amp;language=&amp;filter=&amp;action=translate translation or proofreading] of this banner text.

'''You will be able to read, but not edit, all wikis for a short period of time.'''

* You will not be able to edit for up to an hour on {{#time:l j xg Y|2026-10-28|en}}.
* If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

* Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[user:Trizek_(WMF)|Trizek_(WMF)]] 14:00, 26 October 2026 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&amp;oldid=30513874 --&gt;

== Happy birthday to us ==
I just realized that en-wv was opened August 15, 2006. We've reached the 20 year mark! See [[Wikiversity:History of Wikiversity|History of Wikiversity]]. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 19:01, 28 October 2026 (UTC)

== Blocked users on wikimedia projects ==
During my almost 20 years &quot;editing&quot; on various wikimedia wikis I have seen many users being blocked. Some are blocked for short periods, others indefinitely (or infinitely). In most cases a blocking admin provides a rationale for the blocking such as vandalizing, spamming, paid editing, etc. Some of the reasons for blocking are well documented on the English Wikipedia, but some reasons are obscure. I have not been contributing to this project on a regular basis', so am not sure if this is a topic that should be discussed at all at the English Wikiversity. I hope  that there are others who share my interest. 

I believe that not allowing a discussion of this topic will end up on another website, and will draw attention away from wikimedia which is already suffering the loss of many contributions, human and financial.

Any thoughts?

[[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 18:24, 4 November 2026 (UTC) some minor changes [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 02:15, 5 November 2026 (UTC)

== New vote for approval of inactivity policy at its talk page ==
I started a vote on approving the inactivity policy, please vote at [[Wikiversity talk:Inactivity policy#Approval]] instead of here. Thank you. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 17:05, 6 November 2026 (UTC)

== Commons vote about watermarking AI images ==
There is a Wikimedia Commons vote currently taking place about a proposal that would require watermarking of &quot;realistic&quot; uploaded AI images - it closes 13 Nov: [[c:Commons:Requests for comment/Policy update for AI content]]. Consider having your say. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:05, 8 November 2026 (UTC)

:{{comment}} here's the link: [[commons:Commons:Requests for comment/Policy update for AI content|Commons:Requests for comment/Policy update for AI content]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:26, 8 November 2026 (UTC)</text>
      <sha1>l2vvc3t1ivbque01rtso8lx9vawm1uq</sha1>
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  </page>
  <page>
    <title>Wikiversity:Assume good faith</title>
    <ns>4</ns>
    <id>1538</id>
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      <timestamp>2026-11-09T13:22:54Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-59351-26|~2026-59351-26]] ([[User_talk:~2026-59351-26|talk]]) to last version by [[User:YiFeiBot|YiFeiBot]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="2775" sha1="kc29dzcv8k00m6ldrhcal1ju6s5yubt" xml:space="preserve">{{policy|WV:AGF|WV:FAITH}}

To '''assume good faith''' is a fundamental principle on [[Wikiversity]]. As we allow anyone to edit, it follows that we assume that most people who work on the project are trying to help it, not hurt it. If this weren't true, a project like Wikiversity would be doomed from the beginning.  This principle is also called the '''principle of first trust.'''

When you can reasonably assume that a mistake someone made was a well-intentioned attempt to further the goals of the project, correct it without criticizing.  When you disagree with someone, remember that they probably believe that they are helping the project. Consider using [[Wikiversity:Talk page|talk page]]s to explain yourself, and give others the opportunity to do the same. This can avoid misunderstandings and prevent problems from escalating.

Newcomers unaware of Wikiversity's unique culture and the mechanics of Wikiversity editing often make mistakes or fail to respect community norms.  It is not uncommon for a newcomer to believe that an unfamiliar policy should be changed to match their experience elsewhere.  Similarly, many newcomers bring with them experience or expertise for which they expect immediate respect.  Behaviors arising from these perspectives are not malicious.

Obviously, editors can't be expected to assume good faith if, despite the best possible interpretation of another's actions, it is clear that these actions are contrary to the project's goals. Actions inconsistent with good faith contributions to the project's goals include vandalism, [[w:Sockpuppet (internet)|sockpuppetry]], and other clear instances of intentional deceit. 

Assuming good faith also does not mean that no action by editors should be criticized, but instead that criticism should not be attributed to malice unless there is specific evidence of malice. Accusing the other side in a conflict of not assuming good faith, without showing reasonable supporting evidence, is another form of failing to assume good faith.

==See also==
*[[Wikiversity:Please do not bite the newcomers|Please do not bite the newcomers]]

===Essays===
* [[w:Wikipedia:Assume the assumption of good faith|Wikipedia:Assume the assumption of good faith]]
* [[w:Wikipedia:No angry mastodons|Wikipedia:No angry mastodons]]
* [[w:Wikipedia:On assuming good faith|Wikipedia:On assuming good faith]]
* [[w:Wikipedia:Assume bad faith|Wikipedia:Assume bad faith]]
* [[:m:WikipediAhimsa|Wikipedia policy should follow the spirit of ahimsa (from meta)]]
* [[w:Wikipedia:Forgive and forget|Forgive and forget]] - essay at Wikipedia

===Articles===
* [[w:Ethic of reciprocity|Ethic of reciprocity]]
* [[w:Hanlon's razor|Hanlon's razor]]
* [[w:Reciprocal altruism|Reciprocal altruism]]

{{official policies}}</text>
      <sha1>kc29dzcv8k00m6ldrhcal1ju6s5yubt</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Sandbox</title>
    <ns>4</ns>
    <id>1558</id>
    <revision>
      <id>2845040</id>
      <parentid>2844455</parentid>
      <timestamp>2026-11-09T23:59:06Z</timestamp>
      <contributor>
        <username>~2026-59800-88</username>
        <id>3114100</id>
      </contributor>
      <comment>sandbox tutorial</comment>
      <origin>2845040</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="505" sha1="6ra6bufboq7ubus9cm6v8lj9l3ynaos" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}


What
&lt;quiz&gt;
1111111-000000=1111111
90+10=100
&lt;/quiz&gt;
final work

&lt;quiz&gt;
10-2=8
212+8+220
scotland forever
A
A
G
AG
A
KAGA
KG:AS
LAG
SGSA&quot;ASG

G
GSA
AGSA
GS
AG
GA
GS
GSAG
AS
&lt;nowiki&gt;&lt;quiz&gt;&lt;/nowiki&gt;
22-1=21
23+21(11)=354 &lt;======= What does this mean???

i don't know you tell me
EDUCATION = GOOD 
who tells what to tell who

hi im tsg, i love learning!


Sandbox practice edit by Milan Horvat 9: learning how editing works before creating a course page.</text>
      <sha1>6ra6bufboq7ubus9cm6v8lj9l3ynaos</sha1>
    </revision>
    <revision>
      <id>2845041</id>
      <parentid>2845040</parentid>
      <timestamp>2026-11-10T00:00:33Z</timestamp>
      <contributor>
        <username>~2026-59800-88</username>
        <id>3114100</id>
      </contributor>
      <origin>2845041</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="477" sha1="ot2rou0m0temltsp7yf7ut6to7t9fcw" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}


What
&lt;quiz&gt;
1111111-000000=1111111
90+10=100
&lt;/quiz&gt;
final work

&lt;quiz&gt;
10-2=8
212+8+220
scotland forever
A
A
G
AG
A
KAGA
KG:AS
LAG
SGSA&quot;ASG

G
GSA
AGSA
GS
AG
GA
GS
GSAG
AS
&lt;nowiki&gt;&lt;quiz&gt;&lt;/nowiki&gt;
22-1=21
23+21(11)=354 &lt;======= What does this mean???

i don't know you tell me
EDUCATION = GOOD 
who tells what to tell who


Sandbox practice edit by Milan Horvat 9: learning how editing works before creating a course page.</text>
      <sha1>ot2rou0m0temltsp7yf7ut6to7t9fcw</sha1>
    </revision>
  </page>
  <page>
    <title>Interior design</title>
    <ns>0</ns>
    <id>39566</id>
    <revision>
      <id>2845080</id>
      <parentid>2792436</parentid>
      <timestamp>2026-11-10T06:36:47Z</timestamp>
      <contributor>
        <username>~2026-59627-08</username>
        <id>3114111</id>
      </contributor>
      <comment>/* External links */ Supplement the furniture elements for the interior design.</comment>
      <origin>2845080</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="908" sha1="jlb36ht65byxm47qege2uuad2qlz9do" xml:space="preserve">This area is devoted to learning, teaching, and research related to interior design. &lt;!--House planning and single, double, multi-story building.--&gt;
{{RightTOC}}

==Discussion questions==
* What renowned interior designers have influenced you the most? How so? 

==Learning resources==
{{Wikipedia}}
* [[/Words To Know/]]
* [[/Selecting Furniture and Decorations/]]
* [[Feng Shui]]
* [[Green design]]

==Readings==
===Wikipedia===
* [[w:Interior design|Interior design]]
* [[w:American Society of Interior Designers|American Society of Interior Designers]]
* [[w:Japanese_architecture#Japanese_interior_design|Japanese interior design]]
* [[w:Interior design regulation in the United States|Interior design regulation in the United States]]

==External links==
* [https://lenatees-furniture.com/Furniture Projects]
* [http://sourceforge.net/projects/sweethome3d/ Sweet Home 3D]
[[Category:Interior design| ]]</text>
      <sha1>jlb36ht65byxm47qege2uuad2qlz9do</sha1>
    </revision>
    <revision>
      <id>2845081</id>
      <parentid>2845080</parentid>
      <timestamp>2026-11-10T06:38:45Z</timestamp>
      <contributor>
        <username>~2026-59627-08</username>
        <id>3114111</id>
      </contributor>
      <comment>/* External links */ Correction</comment>
      <origin>2845081</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="909" sha1="0foos4emmipiftjtlfugkij3hxowtkn" xml:space="preserve">This area is devoted to learning, teaching, and research related to interior design. &lt;!--House planning and single, double, multi-story building.--&gt;
{{RightTOC}}

==Discussion questions==
* What renowned interior designers have influenced you the most? How so? 

==Learning resources==
{{Wikipedia}}
* [[/Words To Know/]]
* [[/Selecting Furniture and Decorations/]]
* [[Feng Shui]]
* [[Green design]]

==Readings==
===Wikipedia===
* [[w:Interior design|Interior design]]
* [[w:American Society of Interior Designers|American Society of Interior Designers]]
* [[w:Japanese_architecture#Japanese_interior_design|Japanese interior design]]
* [[w:Interior design regulation in the United States|Interior design regulation in the United States]]

==External links==
* [https://lenatees-furniture.com/ Furniture Projects]
* [http://sourceforge.net/projects/sweethome3d/ Sweet Home 3D]
[[Category:Interior design| ]]</text>
      <sha1>0foos4emmipiftjtlfugkij3hxowtkn</sha1>
    </revision>
  </page>
  <page>
    <title>Microsoft Windows/Windows XP</title>
    <ns>0</ns>
    <id>107333</id>
    <revision>
      <id>2845051</id>
      <parentid>2478533</parentid>
      <timestamp>2026-11-10T02:42:17Z</timestamp>
      <contributor>
        <username>~2026-59547-12</username>
        <id>3114105</id>
      </contributor>
      <comment>Windows XP</comment>
      <origin>2845051</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1459" sha1="nv5d4xvlj69og9ethxb8mbx7y4gqseg" xml:space="preserve">'''&lt;u&gt;Windows Experience (XP)&lt;/u&gt;'''

&lt;u&gt;Windows XP, has many editions Professional and Home Edition are the main ones.  It was released in October 2001.&lt;/u&gt;

'''&lt;u&gt;Built-in User Accounts&lt;/u&gt;'''

&lt;u&gt;Windows XP Home has three built-in user accounts; Administrator, Guest, and SUPPORT_388945a0.
Windows XP Professional has three built-in user accounts; Administrator, Guest, and SUPPORT_388945a0.
Home Edition's Administrator account is defaultly disabled in the local policy which cannot be changed, you can only access it in Safe Mode.
Home Edition's SUPPORT_388945a0 account; is defaultly disabled in the local policy and cannot be access.
Professional's user accounts has full access.&lt;/u&gt;

&lt;u&gt;'''Local Security Policy'''
Accounts: Administrator Account Status; is the security policy that Home edition has that restricts the administrator account. It is defaultly disabled.  Local Security Policy is only in Professional.  You cannot add it.
Local Security Policy (secpol.msc) is in Windows XP Professional.&lt;/u&gt;  

&lt;u&gt;'''Active Directory'''
Active Directory is used in Windows Server 2003, it is for users and computers on a network.  Most schools use active directory for students and staff.  Active Directory (AD), can have Group Policies add for groups of users.  There is [[../Active Directory/]] in schools.  Windows XP Professional and Windows Server 2003 are on the OS built platform.&lt;/u&gt;
[[Category:Operating systems]]
[[Category:Microsoft Windows]]</text>
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  </page>
  <page>
    <title>Cricket</title>
    <ns>0</ns>
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    <revision>
      <id>2844968</id>
      <parentid>2844931</parentid>
      <timestamp>2026-11-09T12:46:35Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-59426-49|~2026-59426-49]] ([[User_talk:~2026-59426-49|talk]]) to last version by [[User:Jtneill|Jtneill]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="1243" sha1="aiikjku70i35zrrmuu6z579u6yrhtk0" xml:space="preserve">{{sports}}
{{0% done}}
[[File:Pollock to Hussey.jpg|thumbnail|right|A bowler, bowling to a batsman.]]

'''[[w:Cricket|Cricket]]''' is an English sport which is very popular in [[Bangladesh]], [[Pakistan]], [[India]], [[Sri Lanka]], and many other countries of the world. Cricket is played by two teams, and each team comprises of eleven players. There are strict rules and regulations which govern the domestic as well as international cricket. Cricket is playing very important roles in promoting trade and commerce both domestically and internationally. Cricket also serves as a means of diplomacy between various countries. Thus, cricket joins cricketing countries together to promote political ties also.
==Scoring==
Cricket scores are recorded on a scorecard, where each batsman's runs, balls faced and manner of dismissal are noted in turn and a side's total is written as runs for the wickets lost, so that a total shown as 112/3 is read as 112 runs for three wickets.&lt;ref&gt;{{cite web |url=https://reddy-books.com/blog/how-to-read-a-cricket-scorecard/ |title=How to Read a Cricket Scorecard: A Complete Guide |website=Reddy Books |access-date=2026-11-08 |language=en-IN}}&lt;/ref&gt;

==Subpages==
{{stub}}

{{wikibooks}}

[[Category:Cricket]]</text>
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      <id>2844973</id>
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      <timestamp>2026-11-09T12:51:54Z</timestamp>
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        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>Undid revision [[Special:Diff/2844892|2844892]] by [[Special:Contributions/~2026-59632-72|~2026-59632-72]] ([[User talk:~2026-59632-72|talk]])</comment>
      <origin>2844973</origin>
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{{0% done}}
[[File:Pollock to Hussey.jpg|thumbnail|right|A bowler, bowling to a batsman.]]

'''[[w:Cricket|Cricket]]''' is an English sport which is very popular in [[Bangladesh]], [[Pakistan]], [[India]], [[Sri Lanka]], and many other countries of the world. Cricket is played by two teams, and each team comprises of eleven players. There are strict rules and regulations which govern the domestic as well as international cricket. Cricket is playing very important roles in promoting trade and commerce both domestically and internationally. Cricket also serves as a means of diplomacy between various countries. Thus, cricket joins cricketing countries together to promote political ties also.
==Subpages==
{{stub}}

{{wikibooks}}

[[Category:Cricket]]</text>
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    </revision>
  </page>
  <page>
    <title>Progress and Prospects in Parkinson's Research/Therapy/Research to Clinic/The route to market</title>
    <ns>0</ns>
    <id>135187</id>
    <revision>
      <id>2845001</id>
      <parentid>2109165</parentid>
      <timestamp>2026-11-09T19:00:22Z</timestamp>
      <contributor>
        <username>~2026-59498-83</username>
        <id>3114089</id>
      </contributor>
      <comment>grammar</comment>
      <origin>2845001</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6896" sha1="pc2kv5obdh5paa19vpcks2injmrtawn" xml:space="preserve">&lt;blockquote&gt;In a 2011 article Michael Kelly described &lt;ref&gt;Kelly, Michael: [[Media:FailedParkinson'sDrugs.pdf|“Why do so many drugs fail at the last hurdle? Is there a serious procedural flaw in drug research?”]], Spring Times, April 2011. [[:Image:FailedParkinson'sDrugs.pdf|File details:]]&lt;/ref&gt; how few Parkinson’s drugs make it through the clinical trials pipeline. Arguably all that the last ten years have brought is improvements to existing treatments, this despite all the breakthroughs in understanding and possible treatment avenues that have opened up. It is salutary to reflect that levodopa, often referred to as the gold standard treatment for Parkinson’s, has been available for forty years.&lt;/blockquote&gt;

===Urgency===
Delays in getting new drugs to market are particularly relevant to people with Parkinson’s. The degenerative nature of the condition means that every year that passes without effective treatments to stop or slow the progression is a year lost. In time, as people with the condition can attest, many of today’s drugs become part of the problem rather than part of the solution. In the opening session of the Worldwide Parkinson’s research congress in Glasgow in 2010 Bryn Williams spoke about ‘the fierce urgency of now’ in the search for a cure.&lt;ref&gt;Williams, Bryn: Opening ceremony, World Parkinson Congress, Glasgow, September 2010.&lt;/ref&gt; Where will this urgency come from?

===Action for improving the process===
Drug repositioning – the use of drugs developed for one purpose being used to treat another – is one opportunity. An example is Exendin-4, a drug developed for the treatment of diabetes that is currently being tested as a Parkinson’s medication. The fact that a drug has already been through the clinical trials process means that it can be tested relatively easily as a treatment for another condition and, if successful, deployed comparatively quickly. Exciting though this is, it may be unrealistic to expect that existing drugs will be able to modify cells in the complex way that the latest research suggests will be necessary.
 
Although no-one wants to take risks with unknown drugs, there is a growing feeling that the current clinical trials procedures are overly bureaucratic and can actually prevent the development of new drugs.&lt;ref&gt;Adams, Stephen: “EU Red Tape Making Drugs trials impossible” Daily Telegraph, February 13th, 2012.&lt;/ref&gt; Pharmaceutical companies may conclude that there is no point devoting the time and money to clinical trials when they have to wait such a long time to receive any payback. There are signs that the problem has at least been recognised. Cancer Research UK has pointed out that since the Clinical Trials Directive came into effect in 2004 the UK’s share of trials across the world has dropped from 6% to 2%. The fact that Pfizer has closed its research and development facility in Kent is another illustration of the problem, and the sort of thing that makes Government take notice. The pharmaceuticals industry is very important to the UK economy and anything that drives work away is to be taken seriously.

The response in the Strategy for UK Life Sciences&lt;ref&gt;“Strategy for UK Life Sciences” published by the Department for Business Innovation and Skills, December 2011.
&lt;/ref&gt; is encouraging. One of the key actions is for the Medicines and Healthcare products Regulatory Agency (MHRA)&lt;ref&gt;MHRA website: http://www.mhra.gov.uk/Howweregulate/Medicines/MISGNewTechnologiesAdvisoryPanel/Earlieraccesstonewmedicinesintheuk/index.htm &lt;/ref&gt; to “bring forward proposals for a new ‘Early Access Scheme’ to increase the speed and efficiency of routes to market approval for innovative, breakthrough therapies.” This initiative is to be welcomed, and could potentially offer access to new drugs before they have been formally licensed. Unfortunately the scheme that was proposed is rather unambitious, envisaging that only one or two medicines might qualify each year. Furthermore, it would only reduce the time to market by about a year. A public consultation on this proposal has taken place and the next steps are awaited. A panel of experts has been tasked to come up with recommendations.

In the US the Food and Drug Administration (FDA) regulates the testing of new medicines and the same criticisms apply to the time taken to get drugs to market. Here a more radical approach has been mooted.&lt;ref&gt;Madden, Bartley: “More Choices, Better Health – Free to choose experimental drugs” published by The Heartland Institute, Chicago, 2008. &lt;/ref&gt; This would involve patients taking a greater risk by using new drugs while they are in development, but only after full disclosure of all the risks and benefits. Here again Parkinson’s patients may be more willing to contemplate this strategy when they weigh up the risks against the knowledge that their condition will deteriorate over time. While this approach does have its merits it would be difficult to implement as it may contravene international agreements about the ethics of conducting medical research among human subjects.

What can be done to encourage the development of new drugs and get them to market quickly? It is crucial to take this opportunity to influence the decision makers who are reviewing the clinical trials process. This underlines the importance of the campaigning work that organisations like Parkinson’s UK and the Cure Parkinson’s Trust carry out. Keeping abreast of what is going on in the UK and Europe and lobbying to influence legislation is critically important. The Parkinson’s community is not alone here and can work with other charities that have an interest in speeding up the route to market for new drugs.

===Raising awareness of clinical trials===
In the short term a more achievable objective is to raise awareness of, and participation in, clinical trials. This is something that all the neurological charities that have an interest in research are working on. In this respect a welcome development from the National Institute for Health Research (NIHR) is the Clinical Trials Gateway website.&lt;ref&gt;Clinical Trials Gateway website http://www.ukctg.nihr.ac.uk/default.aspx&lt;/ref&gt; This enables patients to identify trials that are taking place in their area, together with the recruitment criteria, and enables them to make contact with the research team.  Initiatives such as this, together with a simplification of the clinical trials process, should enable more treatments to get to market in less time. That is what we all want.
==Further reading==
Collins, Francis S. “Reengineering Translational Science: The Time Is Right.” Science Translational Medicine 3, no. 90 (July 6, 2011): 90cm17. http://stm.sciencemag.org/content/3/90/90cm17.full
==References==
&lt;references/&gt;

[[Category:Parkinson's disease]]
[[Category:Interdisciplinary studies]]</text>
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      <contributor>
        <username>Young1lim</username>
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      </contributor>
      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20261109.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261109.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

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      <comment>Removing [[:c:File:Baylonianmaps.JPG|Baylonianmaps.JPG]], it has been deleted from Commons by [[:c:User:Ruthven|Ruthven]] because: per [[:c:Commons:Deletion requests/File:Baylonianmaps.JPG|]].</comment>
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      <text bytes="104792" sha1="19ddk4axklz8vx3o4sufxttkoiaxzpw" xml:space="preserve">[[Image:A Man on the Moon, AS11-40-5903.jpg|thumb|right|250px|Astronaut Buzz Aldrin on the moon. Credit: NASA.]]
A systematically organized body of knowledge on a particular subject is often thought of as a science. The collection of such bodies of knowledge also systematically organized likely constitutes '''the sciences'''.

A more archaic meaning is knowledge of any kind whether found through the use of the scientific method or not.

Perhaps nothing symbolizes the sciences more than astronaut Buzz Aldrin, lunar module pilot, walking on the surface of the Moon near the leg of the Lunar Module (LM) &quot;Eagle&quot; during the Apollo 11 extravehicular activity (EVA). Astronaut Neil A. Armstrong, commander, took this photograph with a 70 mm lunar surface camera. While astronauts Armstrong and Aldrin descended in the Lunar Module (LM) &quot;Eagle&quot; to explore the Sea of Tranquility region of the Moon, astronaut Michael Collins, command module pilot, remained with the Command and Service Modules (CSM) &quot;Columbia&quot; in lunar orbit.

The objective of this lecture is to introduce students and others to ''the sciences''. By the end of this lecture, the student or learner will have an introductory understanding of sciences.

This lecture offers a collaborative environment for the creation, sharing, and discussion of [[open educational resources]], [[open research]] and [[open academia]] regarding ''the sciences''. This lecture welcomes learners of all ages. This lecture does not grant any degrees. This lecture strives to be a [[Portal:Learning Projects|learning project]] corresponding to all sciences at accredited educational institutions and any other topics that are of interest to Wikiversity community members. Providing for learning communities to develop, modify and use the materials on Wikiversity, itself constitutes a way in which research included here by the presence of hypotheses could be done as an activity on Wikiversity. This lecture is dynamic and continues to improve.
{{clear}}

==The sciences==
[[Image:The Scientific Universe.png|right|thumb|300px|As an introduction to science, the Scale of the Universe is mapped to the Branches of Science. Credit: [[commons:User:Efbrazil|Eric Fisk]].{{tlx|free media}}]]
'''Def.''' the disciplines or branches of learning, especially those dealing with measurable or systematic principles, the collective disciplines of study or learning acquired through the scientific method are called '''the sciences'''.

'''Def.''' in the definite, a method of discovering knowledge about the natural world based in making falsifiable predictions (hypotheses), testing them empirically, &amp; developing theories that match known data from repeatable physical experimentation is called '''the scientific method'''.

As an introduction to science, the Scale of the Universe is mapped to the Branches of Science. The Scale of the Universe captures the average diameter of key systems. The scale needs to be cubed to make volume; it takes about 60 trillion atoms to make a human cell, 100 trillion cells to make a person, and 108 billion people have lived to build modern society.

Branches of Science captures the major areas of study in science. The images capture the building block for the scientific branch. Math is built up from binary logic, physics is built on atomic scale physical laws, life science is built by cellular behavior calculated through natural selection, social science is built on the human brain, and earth and space science begins with our own planetary body.

Mathematics is groups of logical units, chemistry is groups of atoms following physical laws, functional biology is groups of cells functioning as organisms and ecosystems, and sociology is groups of people governed by their individual psychology.

The sciences are introduced in alphabetical order after a humanities description of the sciences.
{{clear}}

==Human conditions==
{{main|Human conditions}}
It has been argued that the ultimate purpose of science is to make sense of human beings and our nature- for example in his book ''Consilience'', EO Wilson said &quot;The human condition is the most important frontier of the natural sciences.&quot;&lt;ref name=Wilson&gt;Wilson, EO. 1998. ''Consilience: The unity of knowledge''. New York. Alfred A. Knopf. p334&lt;/ref&gt; [[w:Jeremy Griffith|Jeremy Griffith]] supports this view.&lt;ref name=Griffith&gt;Griffith J. 2011. ''What is Science?''. In ''The Book of Real Answers to Everything!'', {{ISBN|9781741290073}}. http://www.worldtransformation.com/what-is-science/ accessed November 20, 2012.&lt;/ref&gt;

==Anthropology==
{{main|Anthropology/Lecture|Anthropology}}
[[Image:1--closeup-of-the-engraving.jpg|thumb|right|250px|This is a close up of lines from the engraving. Credit: Wim Lustenhouwer, VU University Amsterdam.{{tlx|fairuse}}]]
&quot;Some 540,000 years ago, an ancient ancestor of modern humans took a shark tooth and carefully carved a geometric engraving [shown close-up in the image on the right] on a mollusk shell.&quot;&lt;ref name=Viegas&gt;{{ cite book
|author=Jennifer Viegas
|title=Oldest Art Was Carved Onto Shell 540,000 Years Ago
|publisher=Discovery.com
|location=
|date=3 December 2014
|url=http://news.discovery.com/human/evolution/oldest-art-was-carved-onto-shell-540000-years-ago-141203.htm
|accessdate=2014-12-07 }}&lt;/ref&gt;

&quot;The engraving -- the oldest piece of art ever found by at least 300,000 years -- as well as a shell tool were found at a site in what is now Java, Indonesia.&quot;&lt;ref name=Viegas/&gt;

The discovery was made while &quot;studying a fossil freshwater mussel shell assemblage from a site called Trinil in Java. The mussel shells originally were excavated by Eugène Dubois in the 1890s, but have been stored in the Dubois collection of the Naturalis museum in Leiden, The Netherlands. Sediment within the shells enabled them to be dated using both isotopic and luminescence methods.&quot;&lt;ref name=Viegas/&gt;
{{clear}}

==Archaeology==
{{main|Dominant group/Archaeology|Archaeology}}
[[Image:Gurue Mount Murresse.jpg|thumb|right|250px|This diversified and partly man-shaped landscape in northern Mozambique has some visual appeal (being ''beautiful'' or ''picturesque''). Credit: [[w:User:Paulo Oliveira|Paulo Oliveira]].]]
'''Archaeology''' studies human cultures through the recovery, documentation and analysis of material remains and environmental data, including architecture, artifacts, ecofacts, human remains, and landscapes.

Because archaeology employs a wide range of different procedures, it can be considered to be both a [[science]] and a [[humanities|humanity]],&lt;ref name=&quot;Renfrew_Bahn1991&quot;&gt;Renfrew and Bahn (2004 [1991]:13)&lt;/ref&gt;

Archaeology studies human [[history]] from the development of the first [[w:stone tools|stone tools]] in eastern Africa 3.4 million years ago up until recent decades.&lt;ref name=&quot;consumption&quot;&gt;McPherron, S. P., Z. Alemseged, C. W. Marean, J. G. Wynn, D. Reed, D. Geraads, R. Bobe, and H. A. Bearat. 2010. Evidence for stone-tool-assisted consumption of animal tissues before 3.39 million years ago at Dikika, Ethiopia. Nature 466:857-860&lt;/ref&gt; (Archaeology does not include the discipline of [[paleontology]].) It is of most importance for learning about [[w:Prehistory|prehistoric societies]], when there are no written records for historians to study, making up over 99% of total human history, from the [[w:Palaeolithic|Palaeolithic]] until the advent of literacy in any given society.&lt;ref name=&quot;Renfrew_Bahn1991&quot;/&gt;


{{clear}}

==Architecture==
{{main|Architecture}}
[[Image:FallingwaterWright.jpg|right|thumb|300px|The house on falling water is in West Orange, Pennsylvania, USA. Credit: [[commons:User:Serinde|Serinde]].{{tlx|free media}}]]
'''Def.''' the art and science of designing and managing the construction of buildings and other structures, particularly if they are well proportioned and decorated&quot;&lt;ref name=Derby&gt;{{ cite book
|author=Jim Derby
|title=architecture
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=8 January 2014
|url=https://en.wiktionary.org/wiki/architecture
|accessdate=2017-11-04 }}&lt;/ref&gt; is called '''architecture'''.
{{clear}}

==Astronomy==
{{main|Keynote lectures/Astronomy}}
[[Image:Mira the star-by Nasa alt crop.jpg|thumb|center|350px|This ultraviolet-wavelength image mosaic, taken by NASA's Galaxy Evolution Explorer (GALEX), shows a comet-like &quot;tail&quot; stretching 13 light years across space. Credit: NASA.]]
A '''nomy''' (Latin ''nomia'') is a &quot;system of [[Laws|law]]s governing or [the] sum of [[knowledge]] regarding a (specified) field.&quot;&lt;ref name=Gove&gt;{{ cite book
|author=
|title=Webster's Seventh New Collegiate Dictionary
|publisher=G. &amp; C. Merriam Company
|location=Springfield, Massachusetts
|date=1963
|editor=Philip B. Gove
|pages=1221
|bibcode=
|doi=
|pmid=
|isbn=
}}&lt;/ref&gt; ''Nomology'' is the &quot;[[What is science?|science]] of physical and logical laws.&quot;&lt;ref name=Gove/&gt; When any effort to acquire a system of laws or knowledge focusing on an ''astr'', ''aster'', or ''astro'', that is, any natural body in the sky especially at night,&lt;ref name=Gove/&gt; succeeds even in its smallest [[measurement]], '''[[Keynote lectures/Astronomy|astronomy]]''' is the name of the effort and the result.

==Atmospheric sciences==
{{main|Atmospheres/Sciences|Atmospheric sciences}}
[[Image:Parma 2003-10-24 0350Z.jpg|thumb|right|250px|The Moderate Resolution Imaging Spectroradiometer (MODIS) instrument captured this true-color image of Typhoon Parma on October 24, 2003 at 3:50 UTC. Credit: Jeff Schmaltz, MODIS Rapid Response Team, NASA/GSFC.]]
'''[[Atmospheric sciences]]''' concerns the study of the Earth's atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. It also focuses on the atmospheres of astronomical objects.

'''Def.''' the gases surrounding any astronomical body are called an '''atmosphere'''.

'''Def.''' the study of the atmosphere and its phenomena, especially with weather and weather forecasting is called '''meteorology'''.

'''Def.''' precipitation products of the [[condensation]] of atmospheric water vapour are called '''hydrometeors'''.

&quot;Condensation or sublimation of atmospheric water vapor produces a hydrometeor. It forms in the free atmosphere, or at the earth's surface, and includes frozen water lifted by the wind. Hydrometeors which can cause a surface visibility reduction, generally fall into one of the following two categories:
# '''Precipitation'''. Precipitation includes all forms of water particles, both liquid and solid, which fall from the atmosphere and reach the ground; these include: liquid precipitation (drizzle and rain), freezing precipitation (freezing drizzle and freezing rain), and solid (frozen) precipitation (ice pellets, hail, snow, snow pellets, snow grains, and ice crystals).
# '''Suspended (Liquid or Solid) Water Particles'''. Liquid or solid water particles that form and remain suspended in the air (damp haze, cloud, fog, ice fog, and mist), as well as liquid or solid water particles that are lifted by the wind from the earth’s surface (drifting snow, blowing snow, blowing spray) cause restrictions to visibility. One of the more unusual causes of reduced visibility due to suspended water/ice particles is whiteout, while the most common cause is fog.&quot;&lt;ref name=Mireles&gt;{{ cite book
|author=Mark R. Mireles
|author2=Kirth L. Pederson
|author3=Charles H. Elford
|title=Meteorologial Techniques
|publisher=Air Force Weather Agency/DNT
|location=Offutt Air Force Base, Nebraska, USA
|date=February 21, 2007
|editor=
|pages=
|url=http://oai.dtic.mil/oai/oai?verb=getRecord&amp;metadataPrefix=html&amp;identifier=ADA466107
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=
|accessdate=2013-02-17 }}&lt;/ref&gt;
{{clear}}

==Biology==
{{main|Biology}}
[[Image:0d1d6c a691462cb72584922481fb68974c1bd4.png srz 251 340 75 22 0.50 1.20 0.00 png srz.png|thumb|right|250px|This is a biology collage. Credit: DL Heggie.]]
'''Def.''' a status given to any entity, having the properties of replication and metabolism is called '''life'''.

'''Def.''' the study of all life or living matter including the structure, function, and behavior of an organism or type of organism is called '''biology'''.

[[Botany]] is the study of plants, including their physiology, structure, genetics, ecology, distribution, classification, and economic importance.

'''Def.''' that part of [[biology]] which relates to the [[School:Zoology|animal]] kingdom, including the structure, [[embryology]], [[evolution]], [[classification]], habits, and distribution of all animals, both living and [[extinct]] is called '''[[zoology]]'''.
{{clear}}

==Chemistry==
{{main|Chemicals/Chemistry|Chemistry}}
[[Image:Chemicals in flasks.jpg|thumb|right|250px|Chemicals in flasks (including Ammonium hydroxide and Nitric acid) lit in different colours. Credit: [http://flickr.com/photos/27304596@N00 Joe Sullivan].]]
'''Def.''' a natural science that deals with the composition and constitution of [[substances]] and the changes that they undergo as a consequence of alterations in the constitution of their [[molecules]] is called '''chemistry'''.

'''Def.''' a science that deals with
# the identification of the substances of which matter is composed,
# the investigation of their properties and the ways in which they interact, combine, and change, and
# the use of these processes to form new substances
is called '''chemistry'''.

'''Def.''' &quot;a science that deals with the composition, structure, and properties of substances and of the transformations that they undergo&quot;&lt;ref name=Gove&gt;{{ cite book
|author=
|title=Webster's Seventh New Collegiate Dictionary
|publisher=G. &amp; C. Merriam Company
|location=Springfield, Massachusetts
|date=1963
|editor=Philip B. Gove
|pages=1221
|bibcode=
|doi=
|pmid=
|isbn=
}}&lt;/ref&gt; is called '''chemistry'''.

'''Chemistry''' is the science of [[matter]], especially its [[chemical reaction]]s, but also its composition, structure and properties.&lt;ref name=definition&gt;{{ cite book
|url=http://chemweb.ucc.ie/what_is_chemistry.htm
|title=What is Chemistry?
|publisher=Chemweb.ucc.ie
|date=
|accessdate=2011-06-12 }}&lt;/ref&gt;&lt;ref name=Webster&gt;[http://dictionary.reference.com/browse/Chemistry Chemistry]. (n.d.). Merriam-Webster's Medical Dictionary. Retrieved August 19, 2007.&lt;/ref&gt;
{{clear}}

==Earth sciences==
{{main|Sciences/Earth|:Category:Earth sciences}}
[[Image:Volcano q.jpg|thumb|250px|right|A volcanic eruption is the release of energy from below the surface of Earth. Credit: National Parks Service, USA.]]
'''Earth science''' is an all-embracing term for the [[What is science?|science]]s related to the planet [[Earth]].&lt;ref&gt;{{ cite book
|url=http://www.memidex.com/earth-science
|title=earth science
|work=Memidex/WordNet Dictionary
|accessdate=2012-06-11 }}&lt;/ref&gt;

'''Def.''' the intellectual and practical activity encompassing the systematic study through observation and experiment of the Earth's physical structure and substance, its history and origin, and the processes that act on it, especially by examination of its rocks, is called '''geology'''.

The [[Earth]] is a rocky object whose surface is mostly covered with water that is in orbit around a star called the [[Stars/Sun|Sun]].
{{clear}}

==Geography==
{{main|Geography}}
[[Image:World map 2004 CIA large 1.7m whitespace removed.jpg|thumb|250px|right|A Robinson projection map of the [[Earth]] is displayed in the diagram. Credit: United States Central Intelligence Agency's World Factbook.]]
'''Geography''' is the science and art of locating people and features of interest on the [[Earth]].
{{clear}}

==Geology==
{{main|Geology}}
[[Image:San Andreas.jpg|thumb|right|250px|This aerial view shows a portion of the San Andreas fault in California. Credit: Robert E. Wallace, USGS USA.]]
At its earliest, '''geology''' is the study of the [[rocks]] that compose the astronomical rocky object referred to as the [[Earth]].
{{clear}}

==Logic==
{{main|Logic}}
[[Image:Christine Ladd-Franklin (1847-1930).jpg|thumb|right|250px|In 1869, as women were not granted much access to laboratories and observatories, Christine Ladd-Franklin turned to mathematics and logic, which did not require any apparatus. Credit: [http://www.flickr.com/photos/25053835@N03 Smithsonian Institution]. ]]
'''Logic''' is more than [[reasoning]]. Usually it is reasoning conducted or assessed according to strict principles of [[validity]].

Aristotelian logic is a particular system or codification of the principles of proof and inference.

At a secondary level an [[Introduction to logic|introduction to logic]] may be helpful, where some of the more common operators are described. This introduction is a part of [[Elementary logic|elementary logic]] at the undergraduate level. Here, there is at least one [[Ideas in Geometry/Logic|lesson]] available.

This learning resource is partly an article, in some subareas an essay, and mostly a lecture.

Logic is often considered a part of philosophy. And, most often is used in science to help create knowledge consisting of facts and truths. But, it finds needed applicability in law and the practice of law. A third popular field that confers a rigid structure on logic is mathematics.

Nearly all efforts, intellectual or otherwise, can be approached and have some understanding produced through the application of logic. This includes volition (e.g., emotion), affections, morality, and religion.
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==Materials sciences==
{{main|Chemicals/Materials|Dominant group/Materials science|Materials Science and Engineering}}
[[Image:Clinochlore-usa12c.jpg|thumb|right|250px|Here in this specimen of Tilly material, clinochlore is definitely the dominant species with this freestanding crystal. Credit: [[commons:User:Rob Lavinsky|Rob Lavinsky]], [http://www.irocks.com iRocks.com].]]
'''Materials science''' is an interdisciplinary field applying the properties of matter to various areas of science and engineering.

'''Def.''' an understanding of the nature of substances whose properties make them useful in structures, machines, devices, or products, leading to theories or descriptions that explain how structure relates to composition, properties, and behavior is called '''materials science'''.
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==Mathematics==
{{main|Mathematics}}
[[Image:Superclusters atlasoftheuniverse.gif|thumb|250px|right|The universe within 1 billion light-years (307 Mpc) of Earth is shown to contain the local superclusters, galaxy filaments and voids. Credit: Richard Powell.]]
'''Mathematics''' is about numbers (counting), quantity, and coordinates.
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==Natural sciences==
{{main|Sciences/Naturals|Natural sciences}}
&quot;The '''natural sciences''' are those branches of science that seek to elucidate the rules that govern the natural world through scientific methods.&lt;ref name=Ledoux&gt;{{ cite journal
|title=Defining Natural Sciences
|journal=Behaviorology Today
|date=2002
|author=Stephen F. Ledoux
|volume=5
|issue=1
|pages=34
|url=http://behaviorology.org/pdf/DefineNatlSciences.pdf
|format=PDF }}&lt;/ref&gt;

&quot;Fundamentally, natural sciences are deﬁned as disciplines that deal only with natural events (i.e., independent and dependent variables in nature) using scientiﬁc methods.&quot;&lt;ref name=Ledoux/&gt;

The term &quot;natural science&quot; is used to distinguish the subject from the [[Portal:Social Sciences|social sciences]], which apply the scientific method to study human behavior and social patterns; the [[humanities]], which use a critical or analytical approach to study the human condition; and the formal sciences such as [[mathematics]] and [[logic]], which use an [[w:a priori and a posteriori|a priori]], as opposed to factual methodology to study formal systems.

==Oceanography==
{{main|Liquids/Liquid objects/Oceans/Oceanography|Oceanography}}
[[Image:World11.jpg|thumb|250px|right|This image shows the Earth's oceans and their bottoms. Credit: NOAA.]]
'''Oceanography''' studies the oceans, marine organisms, ecosystems, ocean currents, waves, the ocean floor, and chemical substances.
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==Physics==
{{main|Physics}}
[[Image:CollageFisica.jpg|thumb|right|250px|This collage of images exhibits a variety of physics subfields. Credit: [[commons:User:Aushulz|Aushulz]].]]
'''Def.''' the nature and properties of &quot;matter and energy and their interactions&quot;&lt;ref name=Gove/&gt; is called '''physics'''.

Many of the early laws begin to make sense and increase understanding of the phenomena observed when coupled to experimental and theoretical studies performed in a laboratory here on Earth under controlled conditions.

'''Standard condition for temperature and pressure''': The most used standards are those of the International Union of Pure and Applied Chemistry (IUPAC) and the National Institute of Standards and Technology (NIST).
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==Psychology==
{{main|Psychology}}
[[Image:Laugh.jpg|thumb|right|250px|Laughter can have positive psychological and physiological effects. Credit: [[User:TillyHarper|TillyHarper]].]]
'''Psychology''' is a scientific study of the human mind and its functions, especially those affecting behavior in a given context. It can refer to the mental characteristics or attitude of a person or group, or the mental and emotional factors governing a situation or activity.
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==Sociology==
{{main|Dominant group/Sociology|Sociology}}
[[Image:Sna large.png|thumb|right|250px|This image represents the interactions among entities that constitute a society studied by sociology. Credit: [[commons:User:DarwinPeacock|DarwinPeacock]].]]
'''Sociology''' is the scientific study of society.
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==Theoretical sciences==

'''Def.''' the practice of training people to obey rules or a code of behavior, using punishment to correct disobedience is called '''discipline'''.

'''Def.''' the controlled behavior resulting from such training as mentioned just above is called '''discipline'''.

'''Def.''' an activity or experience that provides mental or physical training is called a '''discipline'''.

'''Def.''' the disciplines or branches of learning, especially those &quot;dealing with measurable or systematic principles ... the collective [disciplines] of study or learning acquired through the scientific method&quot;&lt;ref name=ScienceWikt&gt;{{ cite book
|author=[[wikt:User:Widsith|Widsith]]
|title=science
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=20 August 2006
|url=http://en.wiktionary.org/wiki/science
|accessdate=2018-03-19 }}&lt;/ref&gt; is called '''science'''.

The best way to ruin a potentially capable scientist is to require discipline.

'''Def.''' the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment is called '''science'''.

'''Def.''' a systematically organized body of knowledge on a particular subject is called '''science'''.

==History of scientific events==
[[Image:Newton's reflecting telescope.jpg|right|thumb|300px|Newton's reflector is the first reflecting telescope built in 1668. Credit: David Brewster.{{tlx|free media}}]]
'''Def.''' a period of time that has already happened, in contrast to the present and the future is called a '''past'''.

'''Def.'''
# an occurrence; something that happens
# a point in spacetime having three spatial coordinates and one temporal coordinate,
# a possible action that the user can perform that is monitored by an application or the operating system, or
# a set of some of the possible outcomes; a subset of the sample space

is called an '''event'''.

'''Def.'''
# the aggregate of past events,
# the branch of knowledge that studies the past; the assessment of notable events,
# a set of events involving an entity,
# a record or narrative description of past events,
# the list of past and continuing medical conditions of an individual or family,
# a record of previous user events, especially of visited web pages in a browser, or
# something that no longer exists or is no longer relevant

is called '''history'''.

The following is a reverse chronology of various science events, structures, instruments and artifacts.

==21st Century==
[[Image:Neusun1 superk1.jpg|thumb|right|300px|This &quot;neutrino image&quot; of the Sun is produced by using the Super-Kamiokande to detect the neutrinos from nuclear fusion coming from the Sun. Credit: R. Svoboda and K. Gordan (LSU).]]
Here on the Earth's surface the ν&lt;sub&gt;e&lt;/sub&gt; flux is about 10&lt;sup&gt;11&lt;/sup&gt; ν&lt;sub&gt;e&lt;/sub&gt; cm&lt;sup&gt;-2&lt;/sup&gt; s&lt;sup&gt;-1&lt;/sup&gt; in the direction of the Sun.&lt;ref name=Bahcall2004&gt;{{ cite book
|author=John N. Bahcall
|title=Solving the Mystery of the Missing Neutrinos
|publisher=Nobel Media AB
|location=
|date=April 28, 2004
|url=http://www.nobelprize.org/nobel_prizes/themes/physics/bahcall/
|accessdate=2014-03-08 }}&lt;/ref&gt;

&quot;The total number of neutrinos of all types agrees with the number predicted by the computer model of the Sun. Electron neutrinos constitute about a third of the total number of neutrinos. [...] The missing neutrinos were actually present, but in the form of the more difficult to detect muon and tau neutrinos.&quot;&lt;ref name=Bahcall2004/&gt;
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==20th Century==
[[Image:X-15 flying.jpg|thumb|right|300px|The image shows a North American X-15 on a test flight for the US Air Force. Credit: USAF.]]
Manned '''spaceflight''' on an individual basis has only been achieved with experimental aircraft such as the X-15.
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==19th Century==
[[Image:Covered Bridge Cedarburg WI May-09.jpg|right|thumb|300px|This Covered Bridge is a latticed timber truss bridge built in 1867 as of May 2009. Credit: [[commons:User:Freekee|Freekee]].{{tlx|free media}}]]
The Covered Bridge on the right was built in 1867 near the town of Cedarburg, Wisconsin.
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==18th Century==
[[Image:Jantar Mantar at Jaipur.jpg|thumb|right|300px|The Jantar Mantar is a collection of architectural astronomical instruments, built by Maharaja Jai Singh II at his then new capital of Jaipur between 1727 and 1733. Credit: [[w:User:Knowledge Seeker|Knowledge Seeker]].]]
Taken from the observation platform at the top of the Jantar Mantar, the image on the right shows smaller architectural sundials. The Jantar Mantar is a collection of architectural astronomical instruments, built by Maharaja Jai Singh II at his then new capital of Jaipur between 1727 and 1733. The City Palace is behind then Govindji Temple. Nahargarh Fort is on the Hill.
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==17th Century==
[[Image:Kepschem.png|thumb|right|300px|This is a schematic of a Keplerian refracting telescope which uses two different sizes of plano-convex lenses. Credit: Szőcs Tamás.]]
The '''Keplerian Telescope''' was invented by Johannes Kepler in 1611.&lt;ref name=Tunnacliffe&gt;{{ cite book
|title= Optics
|author= AH Tunnacliffe, JG Hirst
|date= 1996
|publisher= 
|location= Kent, England 
|isbn= 0-900099-15-1
|pages= 233–7
|url= }}&lt;/ref&gt;
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==16th Century==
[[Image:Armillary sphere, 1547.jpg|thumb|right|250px|This armillary sphere is dated from 1547. Credit: [[commons:User:Mike Peel|Mike Peel]].]]
The images on the right shows the sphere without sighting tubes or any device for observing astronomical objects and dates from 1547.
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==15th Century==
[[Image:Sandro Botticelli 052.jpg|thumb|right|250px|An armillary sphere is shown in a painting by Sandro Botticelli, c. 1480. Credit: [http://mail.wikipedia.org/pipermail/wikide-l/2005-April/012195.html Creator: Sandro Botticelli].]]
The image at the right dating from c. 1480 shows the sphere without sighting tube or any device for observing astronomical objects.
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==14th Century==
[[Image:Remains of a pre-Hispanic temazcal.jpg|right|thumb|300px|The remains of a pre-Hispanic ''temazcal'' recently found in Mexico City helped archaeologists pinpoint the location of the ancient neighborhood of Temazcaltitlan. Credit: Edith Camacho/National Institute of Anthropology and History (INAH).{{tlx|free media}}]]
[[Image:Sceptre de Charles V.jpg|thumb|left|300px|The image shows Charlemagne holding a sceptre dated from before 1380. Credit: [[commons:User:Siren-Com|Siren-Com]].]]
The remains of a pre-Hispanic ''temazcal'' in the image on the right recently found in Mexico City helped archaeologists pinpoint the location of the ancient neighborhood of Temazcaltitlan.

&quot;Mexico City stands on the ancient site of Tenochtitlán, which, by the late 15th century, had emerged as the bustling capital of the Aztec Empire. One of the city’s oldest neighborhoods was Temazcaltitlan, known as a spiritual hub for the worship of female deities.&quot;&lt;ref name=Katz&gt;{{ cite web
|author=Brigit Katz
|title=14th-Century Steam Bath Found in Mexico City
|publisher=Smithsonian Magazine
|location=
|date=24 January 2020
|url=https://www.smithsonianmag.com/smart-news/14th-century-steam-bath-found-mexico-city-180974049/?utm_source=smithsoniandaily&amp;utm_medium=email&amp;utm_campaign=20200124-daily-responsive&amp;spMailingID=41621414&amp;spUserID=OTQ1NTMwNzM4NTY2S0&amp;spJobID=1682260343&amp;spReportId=MTY4MjI2MDM0MwS2
|accessdate=24 January 2020 }}&lt;/ref&gt;

The &quot;''temazcal'', as steam baths are called in the indigenous Nahuatl language, was found near Mexico City’s modern La Merced neighborhood. It is a domed structure, spanning about 16.5 feet long by 10 feet wide, and was made from adobe blocks and stucco-coated tezontle, a type of volcanic rock.&quot;&lt;ref name=Dvorsky&gt;{{ cite web
|author=George Dvorsky
|title=14th-Century Steam Bath Found in Mexico City
|publisher=Smithsonian Magazine
|location=
|date=24 January 2020
|url=https://www.smithsonianmag.com/smart-news/14th-century-steam-bath-found-mexico-city-180974049/?utm_source=smithsoniandaily&amp;utm_medium=email&amp;utm_campaign=20200124-daily-responsive&amp;spMailingID=41621414&amp;spUserID=OTQ1NTMwNzM4NTY2S0&amp;spJobID=1682260343&amp;spReportId=MTY4MjI2MDM0MwS2
|accessdate=24 January 2020 }}&lt;/ref&gt;

&quot;[Y]ou can see the tub or water pool for the steam bath, as well as one of the sidewalks that were part of it.&quot;&lt;ref name=Martes&gt;{{ cite web
|author=Martes
|title=Hallazgo en inmediaciones de La Merced confirma ubicación del barrio prehispánico de Temazcaltitlan
|publisher=Smithsonian Magazine
|location=
|date=21 January 2020
|url=https://www.inah.gob.mx/boletines/8865-hallazgo-en-inmediaciones-de-la-merced-confirma-ubicacion-del-barrio-prehispanico-de-temazcaltitlan
|accessdate=24 January 2020 }}&lt;/ref&gt;

&quot;The area was known for at least one ''temazcal'', mentioned in the ''Crónica Mexicáyotl'' by Hernando Alvarado Tezozómoc, a 16th-century Nahua nobleman who wrote about the ascendance and fall of the Aztec capital. According to ''Tezozómoc'', a ''temazcal'' was built in the area to purify a noble girl named Quetzalmoyahuatzin; the neighborhood got its name, Tezozómoc notes, because “all the Mexicans bathed […] there.”&quot;&lt;ref name=Katz/&gt;

&quot;Temazcaltitlan was associated with the worship of female deities of fertility, water, and ''pulque'', a fermented agave drink with ancient roots; the Aztec goddess Mayahuel is often depicted with agave sap pouring from her breasts. The discovery of the temazcal, experts say, confirms the neighborhood’s status as a spiritual center.&quot;&lt;ref name=Katz/&gt;

The image at the left shows a sceptre dated to before 1380.
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==13th Century==
[[Image:Canyons of the Ancients night ruin.jpg|right|thumb|300px|This Anasazi ruin tower is found in the Canyon of the Ancients, New Mexico. Credit: Bob Wick.{{tlx|free media}}]]
The ruin tower on the right is apparently dated to the 13th Century and was built by the Anasazi.
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==12th Century==
[[Image:Iglesia de San Tirso de Sahagún.jpg|right|thumb|300px|Exterior is of the Church of San Tirso, Sahagún.  Credit: [http://flickr.com/photos/17364971@N00 José Antonio Gil Martínez].{{tlx|free media}}]]
Romanesque apses and brick towers of the Church of San Tirso, Sahagún, are shown on the right, dated to the 12th Century.
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==11th Century==
[[Image:Su Song Star Map 1.JPG|thumb|right|300px|This star map is from Su Song's ''Xin Yi Xiang Fa Yao''. Credit: [[commons:User:PericlesofAthens|PericlesofAthens]].]]
The star map on the right, which features a cylindrical projection, was published in 1092 and has a corrected position for the pole star using Shen Kuo's astronomical observations.&lt;ref name=Needham&gt;{{ cite book
|author=Joseph Needham
|date=1986
|title=Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth
|location=Taipei
|publisher=Caves Books Ltd
|pages=208 }}&lt;/ref&gt;
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==10th Century==
[[Image:Visby 11-.JPG|right|thumb|300px|Photo was taken in Visby in August 2007. Credit: [[commons:User:Żeglarz|Żeglarz]].{{tlx|free media}}]]
Visby was founded in the 10th century, on the then independent Baltic Sea island of Gotland. The Hansaetic League formed it during ensuing centuries, during which it came to Denmark. In 1645, it came into Swedish occupation, in which it has remained until today.

There is more about lenses more recently from Visby, Gotland.

&quot;What intrigues the researchers is that the lenses are of such high quality that they could have been used to make a telescope some 500 years before the first known crude telescopes were constructed in Europe in the last few years of the 16th century.&quot;&lt;ref name=Whitehouse&gt;{{ cite book
|author=David Whitehouse
|title=Did the Vikings make a telescope?
|publisher=BBC News
|location=
|date=April 5, 2000
|url=http://news.bbc.co.uk/2/hi/science/nature/702478.stm
|accessdate=2012-10-03 }}&lt;/ref&gt;

&quot;Made from rock-crystal, the lenses have an accurate shape that betrays the work of a master craftsman. The best example of the lenses measures 50 mm (2 inches) in diameter and 30 mm (1 inch) thick at its centre.&quot;&lt;ref name=Whitehouse/&gt;

&quot;The [Visby] Gotland crystals provide the first evidence that sophisticated lens-making techniques were being used by craftsmen over a 1,000 years ago.&quot;&lt;ref name=Whitehouse/&gt;

==9th Century==
[[Image:Cairo Nilometer 1.jpg|right|thumb|300px|Nilometer is at Rawda, Cairo, Egypt. Credit: [[commons:User:Baldiri|Baldiri]].{{tlx|free media}}]]
[[Image:Climbing pyram.jpg|left|thumb|300px| Credit: [[commons:User:Leon petrosyan|Leon petrosyan]].{{tlx|free media}}]]
There was an inscription which placed the foundation of the nilometer in 861.

Cobá is a former pre-Columbian Mayan city on the Yucatán Peninsula southeast of Valladolid located in the state of Quintana Roo, Mexico.

In Cobá, the temple pyramid Nohoch-Mul (also known as Castillo, or the climbing pyramid shown on the left) is 42 meters high.

The city was founded shortly after the beginning of the year and expanded into a city state that peaked between 600 and 800 (1400 and 1200 b2k).
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==8th Century==
[[Image:Dunhuang Star Atlas - complete.jpg|thumb|center|400px|The Dunhuang Star Atlas is the last section of Or.8210/S.3326. Credit: Unknown.]]
[[Image:Dunhuang star map.jpg|thumb|250px|This is an image of the [[w:Dunhuang map|Dunhuang map]] from the [[w:Tang Dynasty|Tang Dynasty]] of the North Polar region. Constellations of the three schools are distinguished with different colors: white, black and yellow for stars of [[w:Wu Xian (astronomer)|Wu Xian]], [[w:Gan De|Gan De]] and [[w:Shi Shen|Shi Shen]] respectively. Credit: [[commons:User:Laurascudder|Laurascudder]], from: Brian J. Ford (1993). Images of Science: A History of Scientific Illustration, Oxford University Press. {{ISBN|0195209834}}.]]
The [[w:Dunhuang map|Dunhuang map]] from the [[w:Tang Dynasty|Tang Dynasty]] of the North Polar region at right is thought to date from the reign of [[w:Emperor Zhongzong of Tang|Emperor Zhongzong of Tang]] (705–710). Constellations of the three schools are distinguished with different colors: white, black and yellow for stars of [[w:Wu Xian (astronomer)|Wu Xian]], [[w:Gan De|Gan De]] and [[w:Shi Shen|Shi Shen]] respectively. The whole set of star maps contains 1,300 stars.

The Dunhuang Star Atlas, the last section of manuscript Or.8210/S.3326. It is &quot;the oldest manuscript star atlas known today from any civilisation, probably dating from around AD 700. It shows a complete representation of the Chinese sky in 13 charts with over 1300 stars named and accurately presented.&quot;&lt;ref name=BritishLibrary&gt;{{ cite book
|author=British Library
|title=The Dunhuang Star Atlas
|publisher=International Dunhuang Project (IDP)
|location=British Library
|date=June 2015
|url=http://idp.bl.uk/4DCGI/education/astronomy/atlas.html
|accessdate=2015-12-27 }}&lt;/ref&gt;

&quot;The Dunhuang Star Atlas [above center] forms the second part of a longer scroll (Or.8210/S.3326) that measures 210 cm long by 24.4 cm wide and is made of fine paper in thirteen separate panels.&quot;&lt;ref name=BritishLibrary/&gt;

&quot;The first part of the scroll is a manual for divination based on the shape of clouds. The twelve charts showing different sections of the sky follow these. The stars are named and there is also explanatory text. The final chart is of the north-polar region. The chart is detailed, showing a total of 1345 stars in 257 clearly marked and named asterisms, or constellations, including all twenty-eight mansions.&quot;&lt;ref name=BritishLibrary/&gt;

&quot;The importance of the chart lies in both its accuracy and graphic quality. The chart includes both bright and faint stars, visible to the naked eye from north central China&quot;.&lt;ref name=BritishLibrary/&gt;
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==7th Century==
[[Image:Sambor Prei Kuk N16 A.jpg|right|thumb|300px|N16 Sanctuary is in Sambor Prei Kuk, Cambodia. Credit: [[commons:User:Baldiri|Baldiri]].{{tlx|free media}}]]
[[Image:Cantona (20065785913).jpg|left|thumb|300px|Cantona is a Mesoamerican archaeological site in the state of Puebla, Mexico. Credit: [https://www.flickr.com/people/67769030@N07 Arian Zwegers from Brussels, Belgium].{{tlx|free media}}]]
Sambor Pre Kuk, with its N16 Sanctuary imaged on the right, is an archaeological complex formed by the remains of the city of Isanapura, the capital of the kingdom of Chenla, an immediate predecessor of the Khmer Empire (pre-Angkorian).

This city was built during the reign of Isanavarman I (616-635). At this time, several constructions, clear predecessors of Khmer architecture, were erected in Angkor.

Cantona is a Mesoamerican archaeological site in the state of Puebla, Mexico. It was a fortified city with a high urbanization level at prehispanic times, probably founded by Olmec-Xicalanca groups towards the late Classical Period. It sat astride an old trading route between the Gulf Coast and the Central Highlands and was a prominent, if isolated, Mesoamerican city between 600 and 1000 CE. After Chichimec's invasions in the 11th century, Cantona was abandoned.

Cantona's inhabitants were mainly agricultural farmers and traders, particularly for obsidian, obtained from Oyameles-Zaragoza mountains surrounding the city. Additionally, they may have been supplying the lowlands with a derivative of the maguey plant, pulque. Cantona's population is estimated at about 80,000 inhabitants at its peak.

Cantona may well be the largest prehispanic city yet discovered in Mesoamerica. Limited archaeological work has been done at the site, and only about 10% of the site can be seen. The Pre-Columbian settlement area occupies approximately 12 km², distributed in three units, of which the largest is at the south, with a surface of 5 km². The site comprises a road network with over 500 cobblestone causeways, more than 3,000 individual patios, residences, and 24 ball courts - more than in any other mesoamerican site. It has an elevated Acropolis over the rest of the city in which the main buildings of the city were built. This was used for the ruling elite and priests, and was where the temples of the most important deities where located. These impressive buildings were constructed with carved stones (one atop the other) without any stucco or cement mortar. Cantona certainly was built with a definite urban design and walkways connecting each and every part of the city. The &quot;First Avenue&quot; is 563 meters in length.
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==6th Century==
[[Image:Basilica Cistern in Istanbul, Turkey (6th century, former Constantinople) (32691506623).jpg|right|thumb|300px|Basilica Cistern is in Constantinople, Turkey, dates from the 6th century. Credit: [https://www.flickr.com/people/145472109@N04 Ralf Steinberger from Northern Italy and Berlin].{{tlx|free media}}]]
[[Image:Dzibanche Buhos2.jpg|thumb|left|300px|The Temple of the Owl pyramid is at Dzibanche. Credit: [[commons:User:HJPD|HJPD]].]]
On the right is the Basilica Cistern in Constantinople, Turkey. It has been dated to the 6th century.

Dzibanche is an archaeological site which includes the Temple of the Owl pyramid. It is an ancient Maya site located in southern Quintana Roo, in the Yucatan Peninsula of southeastern Mexico.

Structures at Dzibanche include the Temple of the Captives, the Temple of the Lintels and the Temple of the Owl, on the left.
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==5th Century==
[[Image:Iron Pillar at Qutub Minar Complex India-0346 - Flickr - archer10 (Dennis).jpg|right|thumb|250px|Iron Pillar is seven-metres high in the courtyard of the mosque and has been there long before the mosque's construction. Credit: [http://www.flickr.com/people/22490717@N02 Dennis Jarvis from Halifax, Canada].{{tlx|free media}}]]
Ancient India was an early leader in metallurgy, as evidenced by the wrought-iron Pillar of Delhi in the image on the right, dated to about 415 or 1585 b2k.
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==4th Century==
[[Image:Capernaum, Israel 06.jpg|right|thumb|300px|This shows the House of Peter, Capernaum, Israel. Credit: [[commons:User:Hoshvilim|Hoshvilim]].{{tlx|free media}}]]
The House of Peter in Capernaum, Israel, has been dated to the 4th century.
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==3rd Century==
[[Image:Dermainzer Globus.jpg|thumb|right|250px|The Mainz celestial globe is the last known celestial globe of Roman antiquity (1850-1780 b2k, 11 cm diameter). Credit: Gunnar Heinsohn.]]
[[Image:Der mainzer Globus.jpg|thumb|left|250px|These are constellation illustrations on the last known celestial globe of Roman antiquity. Credit: Gunnar Heinsohn.]]
The last known celestial globe shown at the right dates from 1850 to 1780 b2k. The constellation illustrations from the Mainz celestial globe are shown at the left.
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==2nd Century==
[[Image:P. Oxy. I 29.jpg|right|thumb|300px|Oxyrhynchus papyrus (P.Oxy. I 29) shows a fragment of Euclid's ''Elements''. Credit: Euclid.{{tlx|free media}}]]
On the right is an image of the oldest extant diagram of Euclid's ''Elements'', found at Oxyrhynchus and dated to c. 100 AD.&lt;ref name=Casselman&gt;{{ cite book
|url=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html
|title=One of the Oldest Extant Diagrams from Euclid
|author=Bill Casselman
|publisher=University of British Columbia
|accessdate=26 September 2008 }}&lt;/ref&gt;
{{clear}}

==1st Century==
Pompeii lens (plano-convex) from the excavations of the Via Stabia, &quot;House of the Engraver&quot; dated to 79 AD per E. Gerspach, ''L'art de la verrerie'' (Paris 1885) 41-42.

Roman London lens (fragmentary biconvex glass lens of light green color), from Roman London, dated to 43-50 AD per H. Syer Cuming, &quot;On Spectacles,&quot; The Journal of the British Archaeological Association 11 (1855) 144-49.

==1st Century BC==
[[Image:MauesBuddhaCoin.JPG|thumb|300px|right|Indian-standard coin of King Maues has on the obverse a rejoicing elephant holding a wreath, symbol of victory. Credit: [[w:user:Per Honor et Gloria|Per Honor et Gloria]]].]]
The &quot;Late La Tène time span [is] between the conquests of 55 BC and 54 BC [2055 and 2054 b2k] under Julius Caesar (100-44 BC) and the time of Christ. In the rare cases where pottery and tableware are attributed to Saxons of the 4th/5th c. AD, &quot;astonishingly La Tène art styles [more than 300 years out of fashion] re-emerge as dominant in the northern and western zone.&quot; (Hines 1996, 260)&quot;&lt;ref name=Heinsohn2017&gt;{{ cite journal
|author=Gunnar Heinsohn
|title=ARTHUR OF CAMELOT AND ATHTHE-DOMAROS OF CAMULODUNUM: A STRATIGRAPHY-BASED EQUATION PROVIDING A NEW CHRONOLOGY FOR 1st MIILLENNIUM ENGLAND
|journal=Quantavolution Magazine
|location=
|date=15 June 2017
|url=http://www.q-mag.org/arthur-of-camelot-and-aththe-of-camulodunum.html
|accessdate=2017-06-21 }}&lt;/ref&gt;

&quot;Stamped pottery has had a long and varied history in Britain. There have been periods when it flourished and periods when
it almost totally disappeared. This article considers two variations of the rosette motif (A 5) and their fortunes from the late Iron Age to the Early Saxon period. [...] The La Tène ring stamps [which end in the 1st century BC; GH ] are found in a
range of designs, from the simple negative ring (= AASPS Classification A 1bi) to four concentric negative rings (= AASPS A
2di). These motifs are also found in the early Roman period [1st century AD; GH]. [...] The 'dot rosettes' (= AASPS A 9di)
on bowls from the [Late Latène] Hunsbury hill-fort (Fell 1937) use the same sort of technique as the dimple decoration on
4th-century 'Romano-Saxon' wares.&quot;&lt;ref name=Briscoe&gt;{{ cite journal
|author=D.C. Briscoe
|title=Two Important Stamp Motifs in Roman Britain and Thereafter, In: ''Romano-British Pottery in the Fifth Century''
|journal=Internet Archaeology
|issue=41
|date= 2016
|doi=https://doi.org/10.11141/ia.41.2
|accessdate=2014-10-25 }}&lt;/ref&gt;

In &quot;Šarnjaka kod Šemovca (Dalmatia/Croatia), e.g., contain 700-year-older La Tène and Imperial period items (1st century BC to 3rd
century AD) [...]:&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;A large dugout house (SU 9) was discovered in the course of the investigation in 2006. Its dimensions are 4.8 by 2.1 metres, with a depth of 34 centimetres, and an east-west orientation, deviating slightly along the NE-SW line. It contained numerous sherds of Early Medieval pottery, two fragments of glass, and a small iron spike. Three sherds of Roman pottery [1st-3rd c. CE; GH] and ten sherds of La Tène pottery [ending 1st c. BCE; GH] were also recovered from the house.&quot;&lt;ref name=Bekic&gt;{{ cite journal
|author=L. Bekić
|title=Nalazi 8. i 9. stoljeća sa Šarnjaka kod Šemovca / Finds from the 8th and 9th centuries at Šarnjak near Šemovec
|journal=Vjesnik Arheološkog muzeja u Zagrebu (VAMZ)
|volume=XLIX
|pages=219-248
|date= 2016
|url=
}}&lt;/ref&gt;

&quot;The contemporaneity of Rome’s Imperial period textbook-dated to the 1st-3rd century AD with the Early Middle Ages (8th-10th century AD) is also confirmed for Poland [in the stratigraphic table above]. There, too, Late Latène (conventionally ending 1st c. BC) immediately precedes the Early Medieval period of the 8th-10th c. CE.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;In [the Roman Empire] capital cities, Rome and Constantinople (Heinsohn 2016) [they] build residential quarters, streets, latrines, aqueducts, ports etc. only in one of the three periods—Imperial Antiquity, Late Antiquity, and Early Middle Ages—dated between 1 and 930s AD. In Rome, they are assigned to Imperial Antiquity (1st-3rd c.); in Constantinople, to Late Antiquity (4th-6th c.).&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Roman churches of Late Antiquity and the Early Middle Ages [...] would suffice to confirm the existence of these two periods. The churches are there. However, we never find churches of the 8th or 9th century superimposed on churches of the 4th or 5th century that, in turn, are superimposed on pagan basilicas of the 1st or 2nd century. They all share the same stratigraphic level of the 1st and 2nd/early 3rd century. Moreover, the ground plans of the 4th/5th—as well as the 8th/9th—century churches slavishly repeat the ground plans of 1st/2nd century basilicas, as already pointed out 75 years ago by Richard Krautheimer (1897-1994). It is this period of Imperial Antiquity (with its internal evolution from the 1st to 3rd centuries) that alone builds the residential quarters, latrines, streets, and aqueducts so desperately looked for in Late Antiquity and the Early Middle Ages. Thus, Rome does not have more stratigraphy for the first millennium AD than England or Poland.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Germanic tribes, not only Anglo-Saxons and Frisians but also Franks, had been competing with Rome for the conquest of the British Isles since the 1st century BC&quot;.&lt;ref name=Heinsohn2017/&gt;

&quot;1st century BC &quot;Astonishingly LA TÈNE art styles&quot; (Hines 1996) dominate pottery of SAXONS [and] Powerful LA TÈNE Celts with King Aththe-Domarous of Camulodunum [is the] greatest ruler.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Saxons begin their attack on Britain as early as the 1st century BC. They compete with the Romans, who may have employed Germanic Franks as auxiliary forces. The Saxons invade from the East, i.e., from the German Bight.&quot;&lt;ref name=Heinsohn2017/&gt;

From &quot;the stratigraphy of the Saxon homeland, located around Bremen/Weser inside today’s Lower Saxony [it] is mainly inhabited by Chauci and Bructeri [...] Saxon tribes that are [...] at war with the Romans in the time of Augustus (31 BC-14 AD) and Aththe-Domaros of Camulodunum (Aθθe-Domaros, also read as Addedom-Arus; c. 15-5 BC).&quot;&lt;ref name=Heinsohn2017/&gt;

On the right is an Indian-standard coin of King Maues. On the obverse is a rejoicing elephant holding a wreath, a symbol of victory. The Greek legend reads ΒΑΣΙΛΕΩΣ&amp;#32;ΒΑΣΙΛΕΩΝ&amp;#32;ΜΕΓΑΛΟΥ&amp;#32;ΜΑΥΟΥ (Great King of Kings Maues). The reverse shows the seated king Maues. Kharoshthi legend: RAJATIRAJASA MAHATASA MOASA (Great King of Kings Maues).
{{clear}}

==2nd Century BC==
[[Image:NAMA Machine d'Anticythère 1.jpg|right|thumb|300px|The image shows the main Antikythera mechanism fragment (fragment A). Credit: [[commons:User:Marsyas|Marsyas]].{{tlx|free media}}]]
[[Image:Nike of Samothrake Louvre Ma2369 n4.jpg|right|thumb|250px|The Nike of Samothrace, ca 190 BC, is a masterpiece of Hellenistic art. Credit: Archaeological expedition of Charles Champoiseau, 1863 and 1879.]]
[[Image:Lagid queen Isis Ma3546.jpg|left|thumb|250px|Cleopatra II was born c. 185 BC and died 116 BC. Credit: [[commons:User:Jastrow|Jastrow]].{{tlx|free media}}]]
The &quot;fragments [of the Antikythera Mechanism] contain at least 30 interlocking gear-wheels, along with copious astronomical inscriptions. Before its sojourn on the sea bed, it computed and displayed the movement of the Sun, the Moon and possibly the planets around Earth, and predicted the dates of future eclipses.&quot;&lt;ref name=Marchant&gt;{{ cite journal
|author=Jo Marchant
|title=In search of lost time
|journal=Nature
|date=30 November 2006
|volume=444
|issue=7119
|pages=534-538
|url=https://www.nature.com/nature/journal/v444/n7119/full/444534a.html
|arxiv=
|bibcode=
|doi=10.1038/444534a
|pmid=
|accessdate=2017-11-05 }}&lt;/ref&gt;

The Winged Nike of Samothrace is made from Parian marble, ca. 190 BC? and found in Samothrace in 1863 by the archaeological expedition of Charles Champoiseau, 1863 and 1879.

Cleopatra II on the left was involved in the ruling of Egypt apparently from c. 175 BC to until she died in 116 BC.
{{clear}}

==3rd Century BC==
[[Image:Sea island survey.jpg|right|thumb|250px|Sea Island survey diagram 窥望海岛之图 was first written of by the Chinese mathematician Liu Hui during the Three Kingdoms era (220–280 CE). Credit: Liu Hui.{{tlx|free media}}]]
[[Image:Mexico.Mex.Teotihuacan.PyramidMoon.01.jpg|thumb|right|360px|The Pyramid of the Moon is one of several monuments built in Teotihuacán apparently in the 3rd century BC. Credit: Hajor.{{tlx|free media}}]]
A Sea Island survey diagram shown on the right was first written of by the Chinese mathematician Liu Hui during the Three Kingdoms era (220–280 CE).

&quot;Jastorf (La Tène) culture [3rd to 1st century BC] with bronze and iron technology. Rich building evidence in downtown Bremen.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;About 280 B.C. [2280 b2k], ... Aristarchus of Samos proposed the hypothesis that the Sun is at rest, while the Earth and the planets rotate about the Sun.&quot;&lt;ref name= Waerden &gt;{{ cite journal
|author=B. L. van der Waerden
|title=The Earliest Form of the Epicycle Theory
|journal=Journal for the History of Astronomy
|date= 1974
|volume=5
|issue=
|pages=175-85
|url=http://adsabs.harvard.edu//abs/1974JHA.....5..175V
|arxiv=
|bibcode=1974JHA.....5..175V
|doi=
|pmid=
|accessdate=2011-10-26 }}&lt;/ref&gt; &quot;Aristarchus also figured out how to measure the distances to the Sun and the Moon and their sizes.&quot;&lt;ref name=Koutsoyiannis&gt;{{ cite book
|author=D. Koutsoyiannis
|author2=A. N. Angelakis
|title=Hydrologic and Hydraulic Science and Technology in Ancient Greece, In: ''Encyclopedia of Water Science''
|publisher=Marcel Dekker, Inc.
|location=New York
|date= 2003
|editor=
|pages=415-7
|url=http://www.a-angelakis.gr/files/books/2002Encycl2WatResTechAncGreEntry2.pdf
|bibcode=
|pmid=
|isbn=
|accessdate=2011-10-26 }}&lt;/ref&gt;

The celestial sphere may have been produced very early: According to records, the first celestial globe was made by Geng Shou-chang (耿壽昌) between 70&amp;nbsp;BC and 50&amp;nbsp;BC. In the [[w:Ming Dynasty|Ming Dynasty]], the celestial globe at that time was a huge globe, showing the 28 mansions, celestial equator and ecliptic. None of them have survived.
{{clear}}

==4th Century BC==
[[Image:Dioptra-principe2.jpg|thumb|right|300px|This is a diagram that shows the operational principles of the dioptra. Credit: [[commons:User:Nerijp|Nerijp]].{{tlx|fairuse}}]]
A 3rd&amp;nbsp;century BCE '''dioptra''' in the image at the right is an astronomical and surveying instrument, a sighting tube if fitted with protractors, it could be used to measure angles.
{{clear}}

==5th Century BC==
[[Image:Edzna, Five-Story Building (14364891152).jpg|right|thumb|300px|The Gran Acropolis is in Edzná. Credit: [https://www.flickr.com/people/67769030@N07 Arian Zwegers from Brussels, Belgium].{{tlx|free media}}]]
[[Image:The Parthenon in Athens.jpg|right|thumb|300px|The Parthenon, Athens Greece, photo taken in 1978. Credit: [http://flickr.com/photos/68686051@N00 Steve Swayne] .{{tlx|free media}}]]
Edzná is a Mayan archaeological site in the north of the Mexican state of Campeche.

The Gran Acropolis is on the right.

Edzná was already inhabited in 400 BC (2400 b2k).

Apparently construction on the Parthenon is dated to have begun in 447 BC (2447 b2k) and was completed in 438 BC (2438 b2k) although decoration continued until 432 BC (2432 b2k).
{{clear}}

==6th Century BC==
[[Image:Kroisos stake Louvre G197.jpg|right|thumb|300px|Croesus at the stake is on Side A from an Attic red-figure amphora, ca. 500–490 BC. Credit: Myson.{{tlx|free media}}]]
[[Image:British Museum gold coin of Croesus.jpg|left|thumb|300px|Gold coin of Croesus, the Lydian, around 550 BC, is from modern Turkey. Credit: [[commons:User:BabelStone|BabelStone]].{{tlx|free media}}]]
[[Image:Alkaios Sappho Staatliche Antikensammlungen 2416 n2.jpg|thumb|center|300px|One of the earliest surviving images of Sappho, from c. 470 BC. Credit: Brygos Painter.]]
[[Image:Becan, Structure IX (14364483042).jpg|right|thumb|300px|Becan is an archaeological site of the Maya civilization in pre-Columbian Mesoamerica. Credit: [https://www.flickr.com/people/67769030@N07 Arian Zwegers from Brussels, Belgium].{{tlx|free media}}]]
The map on the right shows Assyria, Babylonia and Armenia.

Croesus, 595 BC – c. 546 BC, was king of Lydia for apparently 14 years: from 560 BC until his defeat by the Persian king Cyrus the Great in 546 BC, or 547 BC. He issued gold-silver alloy coins as on the left.

Based on the pottery shown in the image on the right, Croesus was burned to death.

&quot;Sappho of Eresos&quot; is shown center holding a lyre and plectrum, and turning toward a bearded man with a lyre partially visible on the left.

On the right is Becán, Structure IX from the top of Structure VIII. Becan is an archaeological site of the Maya civilization in pre-Columbian Mesoamerica. Becan is located near the center of the Yucatán Peninsula, in the present-day Mexican state of Campeche. The name Becan was bestowed on the site by archaeologists who rediscovered the site, meaning &quot;ravine or canyon formed by water&quot; in Yukatek Maya, after the site's most prominent and unusual feature, its surrounding ditch.

Archaeological evidence shows that Becan was occupied in the middle Pre-Classic period, about 550 BCE, and grew to a major population and ceremonial center a few hundred years later in the late Preclassic. The population and scale of construction declined in the early classic (c 250 CE), although it was still a significant site, and trade goods from Teotihuacan have been found. A ditch and ramparts were constructed around the site at this time. There is a ditch that runs the circumference of the city which covers approximately 25 hectares. Around 500 the population again increased dramatically and many large new buildings were constructed, mostly in the Rio Bec style of Maya architecture. Construction of major buildings and elite monuments stopped about 830, although ceramic evidence shows that the site continued to be occupied for some time thereafter, although the population went into decline and Becan was probably abandoned by about 1200.
{{clear}}

==7th Century BC==
[[Image:Amulet to ward off plague.jpg|right|thumb|220px|An Amulet to ward off plague is inscribed with a quotation from the Akkadian Erra Epic. Credit: [[commons:User:Fæ|Fæ]].]]
[[Image:Esarhaddon.jpg|right|thumb|250px|Esarhaddon, king of Assyria, has his portrait on a stone stele, dated after 671 BC. Credit: Maur.{{tlx|free media}}]]
[[Image:Black basalt monument of king Esarhaddon. It narrates Esarhaddon's restoration of Babylon. Circa 670 BCE. From Babylon, Mesopotamia, Iraq. The British Museum, London.jpg|left|thumb|250px|Black basalt monument of king Esarhaddon narrates Esarhaddon's restoration of Babylon. Circa 670 BCE. From Babylon, Mesopotamia, Iraq. The British Museum, London. Credit: [[commons:User:Neuroforever|Osama Shukir Muhammed Amin FRCP(Glasg)]].{{tlx|free media}}]]
Amulet in the image on the right dates from 800 BC-612 BC for warding off plague.

Esarhaddon, portrayed on a stone stele, dated after 671 BC on the right, apparently ruled Assyria between 681 – 669 BC.

The black basalt monument of king Esarhaddon on the left narrates Esarhaddon's restoration of Babylon, ca. 670 BCE, from Babylon, Mesopotamia, Iraq, now in The British Museum, London.
{{clear}}

==8th Century BC==
[[Image:Nimrud lens British Museum.jpg|thumb|right|250px|This image is a photo of the [[w:Nimrud lens|Nimrud lens]] in the [[w:British museum|British museum]]. Credit: [[commons:User:Geni|Geni]].]]
[[Image:Nimrud lens edge on.png|left|thumb|250px|On edge image of the Nimrud lens shows its plano- (left edge) convex (right edge) shape. Credit: Dimitris Plantzos.{{tlx|fairuse}}]]
[[Image:Prothesis Dipylon Painter A517.jpg|thumb|right|300px|Dipylon Vase of the late Geometric period, or the beginning of the Archaic period, is dated c. 750 BC. Credit: Dipylon Master.{{tlx|free media}}]]
[[Image:Sargon II and dignitary.jpg|thumb|left|250px|Sargon II, King of Assyria is depicted here with a dignitary. Credit: [[c:User:Jastrow|Jastrow]].{{tlx|free media}}]]
[[Image:Elissa Statut.jpg|center|thumb|250px|This is a statue of Elissa, Queen of Carthage. Credit: [[c:User:Emnamizouni|Emna Mizouni]].{{tlx|free media}}]]
Nimrud lens is plano-convex, Neo-Assyrian, from the North West Palace, Room AB, dated to 750–710 BC, discovered by Austen Henry Layard at the Assyrian palace of Nimrud, in modern-day Iraq.&lt;ref name=Whitehouse1999&gt;{{ cite book
|author=David Whitehouse
|title=World's oldest telescope?
|url=http://news.bbc.co.uk/1/hi/sci/tech/380186.stm
|date=July 1, 1999
|work=BBC News
|accessdate=May 10, 2008 }}&lt;/ref&gt;

The &quot;&quot;Loupe of Sargon,&quot; a piano-convex rock-crystal lentoid [was] excavated by Layard at Nimrud in the 1850s [on the right].&lt;sup&gt;28&lt;/sup&gt; The object is oval (40 x 35 mm) and of uneven thickness (max. th. 22.5 mm) [centered and including the maximum convexness as shown edge-on in the image on the left. Its focal length has been calculated at 112.5 mm. Its nominal magnification is about 2 x but, owing to its imperfect surface, it would be useless as a tool.&quot;&lt;ref name=Plantzos&gt;{{ cite journal
|author=Dimitris Plantzos
|title=Crystals and Lenses in the Graeco-Roman World
|journal=American Journal of Archaeology
|date=July 1997
|volume=101
|issue=3
|pages=451-64
|url=http://www.hist-arch.uoi.gr/prosopiko/plantzos/Crystals.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2011-10-17 }}&lt;/ref&gt;

This Dipylon Vase on the right is apparently from the late Geometric period, or the beginning of the Archaic period, c. 750 BC (2750 b2k). Prothesis scene is exposure of the dead and mourning.

The low-relief from the L wall of the palace of Sargon II at Dur Sharrukin in Assyria (now Khorsabad in Iraq) is apparently dated to c. 716–713 BC. It is the image on the left.

In the center is an image of the upper portion of a statue of Elissa, Queen of Carthage, apparently from the early 8th century or late ninth century.

&quot;The founding of the Phoenician colony of Toscanos would have come a bit later; its earliest levels (Strata I/II) would date from the years 805–780 cal BC.&lt;sup&gt;6&lt;/sup&gt;&quot;&lt;ref name=Aubet2008/&gt;
{{clear}}

==9th Century BC==
[[Image:ShalmaneserIII.jpg|thumb|right|300px|King Jehu of the Kingdom of Israel bows before Shalmaneser III of Assyria. Credit: [[w:user:Chaldean|Chaldean]].]]
[[Image:Sudan Meroe Pyramids 2001.JPG|right|thumb|300px|These are the pyramids of the Kushite rulers at Meroë. Credit: B N Chagny.{{tlx|free media}}]]
[[Image:Archaeological Sites of the Island of Meroe-114973.jpg|left|thumb|300px|Archaeological Sites of the Island of Meroe is in the Sudan. Credit: Ron Van Oers.{{tlx|free media}}]]
[[Image:Sudanske piramide.JPG|left|thumb|300px|This is another Meroe pyramid. Credit: [[commons:user:Jovisha|Jovan Smiljković]].{{tlx|free media}}]]
In 842 BC the Kingdom of Israel and the Phoenician cities sent tributes to Shalmaneser III. The image on the right is from the Black Obelisk of Shalmaneser III and depicts King Jehu of the Kingdom of Israel bowing before Shalmaneser III of Assyria.

Carthage was founded in 814 BC.&lt;ref name=&quot;Moscati&quot;&gt;{{ cite book
|author=Sabatino Moscati
|editor= Sabatino Moscati
|title=Colonization of the Mediterranean, In: ''The Phoenicians''
|url=https://books.google.com/books?id=1EEtmT9Tbj4C
|accessdate=12 February 2013
|date=12 January 2001
|publisher=I.B.Tauris
|isbn=978-1-85043-533-4
|page=48 }}&lt;/ref&gt;

&quot;The recent radiocarbon dates from the earliest levels in Carthage situate the founding of this Tyrian colony in the years 835–800 cal BC,&lt;sup&gt;4﻿﻿&lt;/sup&gt; which coincides with the dates handed down by Flavius Josephus and Timeus for the founding of the city.&quot;&lt;ref name=Aubet2008&gt;{{ cite book
|title=Political and Economic Implications of the New Phoenician Chronologies
|url=http://www.upf.edu/larq/_pdf/AubtCrono.pdf
|publisher=Universidad Pompeu Fabra
|accessdate=24 February 2013
|author=Maria Eugenia Aubet
|pages=179
|date=2008 }}&lt;/ref&gt;

The &quot;dates for the founding of Carthage coincide with dates established for the Phoenician colony of Morro de Mezquitilla, which situate the most ancient occupation levels (Strata B1a and B2) in the years 807–802 cal BC.&lt;sup&gt;5&lt;/sup&gt;&quot;&lt;ref name=Aubet2008/&gt;

The &quot;founding of the first Phoenician colonies at the end of the ninth century coincides with the radio-carbon chronology attributed to the most ancient Phoenician imports recorded in the indigenous Portuguese settlements and in the south of Spain, like Acinipo, Alcáçova de Santarem or Cerro de la Mora.&lt;sup&gt;7&lt;/sup&gt;&quot;&lt;ref name=Aubet2008/&gt;

On the right is an aerial view of the Nubian pyramids at Meroe in 2001, which apparently date from 2800 to 2280 b2k. The pyramid in the center is from this grouping.

On the left is apparently the second set of Nubian pyramids of the Island of Meroe in the Sudan.
{{clear}}

==10th Century BC==
[[Image:Psusennes I mask.jpg|thumb|right|250px|Gold burial mask is for Pharaoh Psusennes I, discovered 1940 by Pierre Montet. Credit: [[commons:User:Locutus_Borg|José-Manuel Benito]].{{tlx|free media}}]]
[[Image:Mask_of_Amenemope1.jpg|left|thumb|250px|Burial mask is for Pharaoh Amenemope, currently in the Cairo Museum. Credit: [http://www.flickr.com/people/10647023@N04/ tutincommon (John Campana)].{{tlx|free media}}]]
The 10th century BC is equivalent to 3,000 to 2,900 b2k.

On the right is the burial mask for the Pharaoh Psusennes I exhumed from Tomb III at Tanis (Nile delta). The material used is gold (different pieces were assembled using nails). The eyebrows and eye shadows are lapis lazuli. The eyes are glass paste. There is a cobra on the forehead. The ritual beard is braided to symbolize the death of the sovereign. It is kept in the Egyptian Museum of Cairo. Dynasty is the XXI, dated to c. 1039 BC-990 BC.

In the left image is the burial mask for pharaoh Amenemope of the 21&lt;sup&gt;st&lt;/sup&gt; Dynasty of Egypt, dated to 1001 – 992 BC or 993 – 984 BC.
{{clear}}

==11th Century BC==
[[Image:Guanche e.jpg|thumb|right|300px|This is one of the Pyramids of Guimar, Tenerife, Canary Islands. Credit: Museo de la Naturaleza y el Hombre, Tenerife.]]
[[Image:The Pyramids of Güímar, Santa Cruz de Tenerife, Canary Islands, Spain - panoramio.jpg|right|thumb|300px|One of the pyramids of Güímar, Santa Cruz de Tenerife, Canary Islands, Spain, is shown. Credit: [http://www.panoramio.com/user/861544?with_photo_id=102838027 trolvag].{{tlx|free media}}]]
[[Image:The Pyramids of Güímar, Santa Cruz de Tenerife, Canary Islands, Spain - panoramio (1).jpg|right|thumb|300px|The image shows many of the structures associated with the pyramids on Tenerife. Credit: [http://www.panoramio.com/user/861544?with_photo_id=102838027 trolvag].{{tlx|free media}}]]
[[Image:The Pyramids of Güímar visitors senter, Santa Cruz de Tenerife, Canary Islands, Spain - panoramio.jpg|right|thumb|300px|The image shows another view of the many of the structures associated with the pyramids on Tenerife. Credit: [http://www.panoramio.com/user/861544?with_photo_id=102838027 trolvag].{{tlx|free media}}]]
[[Image:The Pyramids of Güímar, Santa Cruz de Tenerife, Canary Islands, Spain - panoramio (3).jpg|right|thumb|300px|The image shows  the pyramid associated with many of the structures on Tenerife. Credit: [http://www.panoramio.com/user/861544?with_photo_id=102838027 trolvag].{{tlx|free media}}]]
[[Image:La Palma - Brena Baja - LP-5 - El Guincho pyramid 02 ies.jpg|right|thumb|300px|The image shows a pyramid on La Palma, Canary Islands that has been damaged by current locals. Credit: Frank Vincentz.{{tlx|free media}}]]
[[Image:Guanche g.jpg|thumb|left|300px|The mummy has red hair and other Nordic features. Credit: Museo de la Naturaleza y el Hombre, Tenerife.]]
[[Image:Gaunches2.jpg|thumb|left|300px|This is another mummy of the Guanches with blonde hair. Credit: Museo de la Naturaleza y el Hombre, Tenerife.]]
[[Image:Pyramid number 4.jpg|right|thumb|300px|The image shows pyramid number four of seven that occur on Mauritius. Credit: Rémy De Saint Simon &amp; Selva Gunness.]]
[[Image:Mauritius-Pyramiden-5-4-3.jpg|left|thumb|300px|Pyramid is situated south of road M2 between Plaine Magnien and New Grove on Mauritius. Credit: [[commons:User:Uli sh|Uli sh]].{{tlx|free media}}]]
[[Image:Pyramid on Pico Island Azores.jpg|left|thumb|300px|One of the 140 pyramids is imaged and observed by archeologists in the Madalena area of Pico Island, Azores. Credit: Carolina Matos.]]
The '''iron age history''' period began between 3,200 and 2,100 b2k.

The Guanches are believed to be the original inhabitants of the Canary Islands perhaps as early as 3,000 b2k.

The first image down on the right is one of the Pyramids of Guimar, Canary Islands.

The &quot;Guanches built several small step pyramids on the islands, using the same model as those found in ancient Egypt and in Mesopotamia.&quot;&lt;ref name=Hwr6/&gt;

&quot;The pyramids have an east-west alignment which also indicates that they probably had a religious [or astronomical] purpose, associated with the rise and setting of the sun.&quot;&lt;ref name=Hwr6/&gt;

&quot;Carefully built stairways on the west side of each pyramid lead up to the summit, which in each case has a flat platform covered with gravel, possibly used for religious or ceremonial purposes.&quot;&lt;ref name=Hwr6/&gt;

They &quot;shared a number of cultural characteristics with the ancient Egyptians and [...] their building style appears to have been replicated in South and Central America.&quot;&lt;ref name=Hwr6&gt;{{ cite book
|author=History of White Race - Chapter 6
|title=The Guanches of the Canary Islands
|publisher=Bibliotecapleyades
|chapter=To The Ends Of The Earth - Lost White Migrations
|chapter-url=http://www.white-history.com/hwr6a.htm
|date= 2008
|url=http://www.bibliotecapleyades.net/esp_guanches_1.htm
|accessdate=2016-01-08 }}&lt;/ref&gt;

The mummies on the left have red hair, the third down has blonde, and other Nordic features of the original inhabitants of the Canary Islands.

&quot;An examination of one of the mummies' bodies showed incisions that virtually matched those found in Egyptian mummies, although the string used by the Guanche embalmers to close the wounds was much coarser than would have been used by the Egyptian experts.&quot;&lt;ref name=Hwr6/&gt;

&quot;Like the '''''Celtic Tocharians''''', the finest evidence of what the original Guanche looked like, is in the fortuitous existence of original Guanche mummies, which are on public display in that island group's national museum.&quot;&lt;ref name=Hwr6/&gt;

&quot;As the original inhabitants of the Canary Islands were fair-haired and bearded, it was possible [...] that long before the 15th Century, people of the same stock as those who settled the Canary Islands, also sailed the same route along the Canary Current that took Christopher Columbus to the Americas.&quot;&lt;ref name=Heyerdahl&gt;{{ cite book
|author=Thor Heyerdahl
|title=The Guanches of the Canary Islands
|publisher=Bibliotecapleyades
|chapter=To The Ends Of The Earth - Lost White Migrations
|chapter-url=http://www.white-history.com/hwr6a.htm
|date= 2008
|url=http://www.bibliotecapleyades.net/esp_guanches_1.htm
|accessdate=2016-01-08 }}&lt;/ref&gt;

&quot;According to the Aztec and Olmec (Central American Amerind) legends, their god, Quetzalcoatl, had Nordic features (eyes and hair color) and a beard. This god came from over the sea and taught the Amerinds how to raise corn and build structures.&quot;&lt;ref name=Hwr6/&gt;

&quot;The existence of the red-haired Guanches on the Canary Islands, combined with the red-haired pre-Columbus mummies found in South America and the marked similarity in pyramid building styles, indicate that an over the atlantic people probably used the Canaries Current to cross the Atlantic, most likely between 2000 and 500 BC.&quot;&lt;ref name=Hwr6/&gt;

&quot;The original occupants of the Canary Islands were a native people called the Guanche.  Their mummies have been found with red hair and blond hair. Reportedly, the men were around 7 feet and the women around 6 feet.&quot;&lt;ref name=Day&gt;{{ cite book
|author=Sharon Day
|title=Canary Islands: Oddities and Coincidences?
|publisher=Ghost Hunting Theories
|location=
|date=18 March 2014
|url=http://www.ghosthuntingtheories.com/2014/03/canary-islands-oddities-and-coincidences.html
|accessdate=2016-01-08 }}&lt;/ref&gt;
Massive &quot;walls, hydraulic systems and road systems on the island of Mauritius, [...] are connected with the pyramid complexes that have been rediscovered there in recent months.&quot;&lt;ref name=Gigal/&gt;

&quot;Since all these structures are in the same general area of the seven pyramids that have been discovered on the island, it is reasonable to conclude that everything was probably built by the same civilisation – and that this was done a long time ago.&quot;&lt;ref name=Gigal/&gt;

&quot;As for the structure of the pyramids on this island off the coast of Africa, they are platform pyramids made of volcanic rock with other stones worked into them, similar in design and size to the pyramids on Tenerife, in the Canary Islands.&quot;&lt;ref name=Gigal&gt;{{ cite book
|author=Antoine Gigal
|title=Megalithic walls and hydraulic systems linked with Mauritius pyramids
|publisher=Gigal Research
|location=United Kingdom
|date= 2013
|url=http://www.gigalresearch.com/uk/complexe-ile-maurice.php
|accessdate=2017-05-10 }}&lt;/ref&gt;

&quot;In the vicinity of these seven pyramids, in a zone that is about two square kilometres in size, near the villages of Plaine Magnien, the cemetery and Mahebourg, there are massive rock walls, constructed with the same techniques and from the same material as the pyramids.&quot;&lt;ref name=Gigal/&gt;

&quot;There are also paved roadways that connect the pyramids to other remarkable structures. The roads are perfectly flat and of such design that the trucks that use them every day cause little damage.&quot;&lt;ref name=Gigal/&gt;

&quot;The long-held notion that the pyramids were built only in Egypt and Mexico had to change in the 20th century. It was discovered that stone and brick structures in the shape of the pyramid were built thousands of years ago from Bolivia and Peru through Central American countries of Mexico, El Salvador, Guatemala, Honduras and Belize as well as throughout the islands of the coasts of Africa such as Canary Islands and Mauritius, then Sudan, Southern European countries such as Italy, France and Greece and in the east in China, Korea and Pacific islands.&quot;&lt;ref name=Osmanagich&gt;{{ cite book
|author=Semir Sam Osmanagich
|title=BOSNIAN VALLEY OF THE PYRAMIDS Discovery and Road to Recognition, In: ''The First International Scientific Conference Bosnian Valley of the Pyramids''
|publisher=the “Archaeological Park: Bosnian Pyramid of the Sun” Foundation
|location=Sarajevo, Bosnia-Herzegovina
|date=25 August 2008
|editor=
|pages=1-55
|url=http://piramidasunca.ba/dokumentovano.com/piramida/dokumenti/referati/ICBP-Osmanagich-Paper-February_2009.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=
|accessdate=2017-05-10 }}&lt;/ref&gt;

&quot;In Mauritius the stony soils form an integral part of the industry,&lt;sup&gt;5&lt;/sup&gt; though often stones are so numerous that they are piled into walls and pyramids, and in parts cultivation is possible only by muraille creole-one row of canes, one of stones- a system that has some advantages for moisture conservation in dry areas.&quot;&lt;ref name=Brookfield&gt;{{ cite journal
|author=H. C. Brookfield
|title=Problems of monoculture and diversification in a sugar island: Mauritius
|journal=Economic Geography
|date= 1959
|volume=35
|issue=1
|pages=25-40
|url=http://www.tandfonline.com/doi/abs/10.2307/142076?journalCode=recg20
|arxiv=
|bibcode=
|doi=10.2307/142076?journalCode=recg20
|pmid=
|accessdate=2017-05-10 }}&lt;/ref&gt;

&quot;It goes without saying that the theory that these are merely heaps of stones is totally absurd, especially when you show them to geologists and engineers. For one thing the precision of the corner angles of these monuments [especially shown in the image on the right] seen from space is unmistakable, and there is no way this could be the work of slaves clearing useless stones from the fields and heaping them up, even artistically. The precision of the corners and bases, and the allowances made for the sometimes uneven ground, demand calculations and execution by experienced architects.&quot;&lt;ref name=Gigal/&gt;

The &quot;Madalena pyramidal structures, known by the locals as “maroiços,” are analogous to similar protohistoric structures found in Sicily, North Africa and the Canary islands which are known to have served ritual purposes.&quot;&lt;ref name=Matos&gt;{{ cite book
|author=Carolina Matos
|title=Pico: New archaeological evidence reveals human presence before Portuguese occupation – Azores
|publisher=Portuguese American Journal
|location=
|date=28 August 2013
|url=http://portuguese-american-journal.com/pico-new-archeological-evidence-reveals-human-presence-before-portuguese-occupation-azores/
|accessdate=2017-04-27 }}&lt;/ref&gt;
{{clear}}

==12th Century BC==

Between 1200 BC and 1000 BC diffusion in the understanding of iron metallurgy and use of iron objects was fast and far-flung. Anthony Snodgrass&lt;ref&gt;A.M.Snodgrass (1966), &quot;Arms and Armour of the Greeks&quot;. (Thames &amp; Hudson, London)&lt;/ref&gt;&lt;ref&gt;A. M. Snodgrass (1971), &quot;The Dark Age of Greece&quot; (Edinburgh University Press, Edinburgh).&lt;/ref&gt; suggests that a shortage of tin, as a part of the Bronze Age collapse and trade disruptions in the Mediterranean around 1300&amp;nbsp;BC, forced metalworkers to seek an alternative to bronze. As evidence, many bronze implements were recycled into weapons during that time. More widespread use of iron led to improved steel-making technology at lower cost. Thus, even when tin became available again, iron was cheaper, stronger and lighter, and forged iron implements superseded cast bronze tools permanently.&lt;ref&gt;Theodore Wertime and J. D. Muhly, eds. ''The Coming of the Age of Iron'' (New Haven, 1979).&lt;/ref&gt;

==13th Century BC==

By the Middle Bronze Age increasing numbers of smelted iron objects (distinguishable from meteoric iron by the lack of nickel in the product) appeared in the Middle East, Southeast Asia and South Asia. African sites are turning up dates as early as 1200 BC.&lt;ref name=&quot;millermintz&quot;&gt;Duncan E. Miller and N.J. Van Der Merwe, 'Early Metal Working in Sub Saharan Africa' ''Journal of African History'' 35 (1994) 1–36; Minze Stuiver and N.J. Van Der Merwe, 'Radiocarbon Chronology of the Iron Age in Sub-Saharan Africa' ''Current Anthropology'' 1968.&lt;/ref&gt;&lt;ref&gt;[http://www.homestead.com/wysinger/ironage.html How Old is the Iron Age in Sub-Saharan Africa?]&amp;nbsp;– by Roderick J. McIntosh, Archaeological Institute of America (1999)&lt;/ref&gt;&lt;ref&gt;[http://muse.jhu.edu/demo/history_in_africa/v032/32.1alpern.pdf Iron in Sub-Saharan Africa]&amp;nbsp;– by Stanley B. Alpern (2005)&lt;/ref&gt;

==14th Century BC==

Schliemann lenses (plano-convex), found at Troy, dated to 1334-1184 BC per V. Tolstikov and M. Treister, The Gold of Troy: Searching for Homer's Fabled City (London 1996) nos. 176-216, 230.

==15th Century BC==
[[Image:Thutmose III sphinx E10897-Louvre 042005 06.jpg|thumb|left|250px|Sphinx-lion of Thutmose III is dated to 1479–1425 BC. Credit: Unknown.]]
&quot;Three bossed crystal discs&quot; (plano-convex), from the Late Minoan II-IIIA, Mavrospelio cemetery, dated to 1425-1340 BC per E. J. Forsdyke, &quot;The Mavro Spelio Cemetery at Knossos,&quot; BSA 28 (1926-1927) 243-96, esp. 288.
{{clear}}

==16th Century BC==
[[Image:Sword bronze age (2nd version).jpg|thumb|right|250px|Ceremonial giant dirk is dated to 1500–1300 BC. Credit: [[c:user:Calame|Calame]].]]
[[Image:Caryatid Mirror, about 1540-1296 BC, New Kingdom, Dynasty 18, bronze perhaps with black copper inlay - Cleveland Museum of Art - DSC08779.JPG|thumb|left|250px|Bronze mirror with a female human figure at the base is dated to the Eighteenth dynasty of Egypt (1540–1296 BC). Credit: [[c:user:Daderot|Daderot]].]]
The Late Bronze Ages begin about 3550 b2k and end about 2900 b2k.

The Pátzcuaro Basin is &quot;on the Central Mexican Altiplano [19° 36′ N 101° 39′ W 2,033–3,000 meters above sea level (m asl)] [...] The earliest occupation is indicated by maize pollen in lake cores [sometime between 1690 and 940 B.C. (43, 47, 49)].&quot;&lt;ref name=Fisher2003&gt;{{ cite journal
|author=Christopher T. Fisher
|author2=Helen P. Pollard
|author3=Isabel Israde-Alcántara
|author4=Victor H. Garduño-Monroy
|author5=Subir K. Banerjee
|title=A reexamination of human-induced environmental change within the Lake Pátzcuaro Basin, Michoacán, Mexico
|journal=Proceedings of the National Academy of Sciences USA
|month=April
|year=2003
|volume=100
|issue=8
|pages=4957-4962
|url=http://www.pnas.org/content/100/8/4957.long
|arxiv=
|bibcode=
|doi=10.1073/pnas.0630493100
|pmid=
|accessdate=2018-2-25 }}&lt;/ref&gt;

The &quot;abandonment of lakeshore Swiss pile-dwellings has been dated to around 1520 BC [3520 b2k] (Menotti 2001). [Slightly] &quot;later in time episodes of flood events and lake-level highstand at 3100 BP (1415/1311 2σ cal. BC) and 2800 BP (996/914 2σ cal. BC) have been recently detected in the Southern Alps, in the sediment cores extracted from the Lake Ledro, located in the province of Trento (Joannin et al. 2014).&quot;&lt;ref name=Capuzzo&gt;{{ cite book
|author=Giacomo Capuzzo
|title=SPACE-TEMPORAL ANALYSIS OF RADIOCARBON EVIDENCE AND ASSOCIATED ARCHAEOLOGICAL RECORD: FROM DANUBE TO EBRO RIVERS AND FROM BRONZE TO IRON AGES
|publisher=UNIVERSITAT AUTÒNOMA DE BARCELONA
|location=BARCELONA, Spain
|year=2014
|volume=
|issue=
|pages=416
|url=https://ddd.uab.cat/pub/tesis/2014/hdl_10803_283401/gc1de1.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2017-10-11 }}&lt;/ref&gt;
Radiocarbon &quot;data indicate that the New Kingdom of Egypt started between 1570 and 1544 B.C.E [3570 - 3544 b2k].&quot;&lt;ref name=Ramsey&gt;{{ cite journal
|author=Christopher Bronk Ramsey
|author2=Michael W. Dee
|author3=Joanne M. Rowland
|author4=Thomas F. G. Higham
|author5=Stephen A. Harris
|author6=Fiona Brock
|author7=Anita Quiles
|author8=Eva M. Wild
|author9=Ezra S. Marcus
|author10=Andrew J. Shortland
|title=Radiocarbon-Based Chronology for Dynastic Egypt
|journal=Science
|month=18 June
|year=2010
|volume=328
|issue=5985
|pages=1554-1557
|url=http://science.sciencemag.org/content/328/5985/1554
|arxiv=
|bibcode=
|doi=10.1126/science.1189395
|pmid=
|accessdate=2017-10-11 }}&lt;/ref&gt;

High precision radiocarbon dating of 18 samples from Jericho, including six samples of carbonized cereal from the burnt stratum, gave the age of the strata as 1562 BC, with a margin of error of 38 years [3562 ± 38 b2k].&lt;ref name=Bruins&gt;{{ cite journal
|author=Hendrik Bruins
|author2=Johannes van der Plicht
|title=Tell-es-Sultan (Jericho): Radiocarbon results of short-lived cereal and multiyear charcoal samples from the end of the Middle Bronze Age
|journal=Radiocarbon
|volume=37
|issue=2
|year=1995
|pages=213-220
|url=https://journals.uair.arizona.edu/index.php/radiocarbon/article/viewFile/1666/1670
|accessdate=2017-10-11 }}&lt;/ref&gt;
{{clear}}

==17th Century BC==
[[Image:Solvogn.jpg|thumb|left|250px|The Trundholm Sun chariot pulled by a horse is a sculpture believed to be illustrating an important part of Nordic Bronze Age mythology. It is displayed at the National Museum of Denmark. Credit: [[commons:User:Malene|Malene Thyssen]].]]
[[Image:Solvogn bagside.jpg|thumb|right|250px|The left side of the disk shows no traces of gilding. Credit: [[commons:User:Malene|Malene Thyssen]].]]
[[Image:Fresco of a fisherman, Akrotiri, Greece.jpg|right|thumb|250px|Fresco of ''The Fisherman'' is from Akrotiri, Santorini, Greece at a height of 1.10 m. Credit: Yann Forget.{{tlx|free media}}]]
The Trundholm Sun chariot pulled by a horse is a sculpture imaged at left and right dated by the Nationalmuseet to about 1800 to 1600 BCE (3800 to 3600 b2k).

Evans lenses (plano-convex) are Middle Minoan IIIB, from the Temple Repositories at Knossos, along with a &quot;royal Draught Board&quot; with ivory and crystal inlays backed with silver foil, dated to 1640-1600 BC, Minoan civilization per A. Evans, ''The Palace of Minos'' I (London 1921) 469-72; cf. the eye of the steatite bull's-head rhyton from the Little Palace at Knossos: Evans, ''The Tomb of the Double Axes etc'' (London 1914) 82.

''The Fisherman'' is a Minoan Bronze Age fresco from Akrotiri, on the Aegean island of Santorini (classically Thera), dated to the Neo-Palatial period (c. 1640–1600 BC). The settlement of Akrotiri was buried in volcanic ash (dated by radiocarbon dating to c. 1627 BC [c. 3626 b2k]) by the Minoan eruption on the island, which preserved many Minoan frescoes like this.
{{clear}}

==19th Century BC==

Souckova-Siegolová (2001) shows that iron implements were made in Central Anatolia in very limited quantities around 1800 BC and were in general use by elites, though not by commoners, during the New Hittite Empire (∼1400–1200 BC).&lt;ref&gt;{{cite journal |last=Souckova-Siegolová |first=J.|title=Treatment and usage of iron in the Hittite empire in the 2nd millennium BC|journal=Mediterranean Archaeology |volume=14 |pages=189–93 |year=2001}}.&lt;/ref&gt;

Similarly, recent archaeological remains of iron working in the Ganges Valley in India have been tentatively dated to 1800&amp;nbsp;BC. Tewari (2003) concludes that &quot;knowledge of iron smelting and manufacturing of iron artifacts was well known in the Eastern Vindhyas and iron had been in use in the Central Ganga Plain, at least from the early second millennium BC&quot;.&lt;ref name=Tewari&gt;{{cite journal
|last=Tewari
|first=Rakesh
|title=The origins of Iron Working in India: New evidence from the Central Ganga plain and the Eastern Vindhyas
|journal=Antiquity
|volume=77
|pages=536–545
|year=2003
|url=http://antiquity.ac.uk/projgall/tewari/tewari.pdf
}}&lt;/ref&gt;

==20th Century BC==

The Iron Age in the Ancient Near East became known with the discovery of iron smelting and smithing techniques in Anatolia or the Caucasus and Balkans in the late 2nd millennium BC ({{circa}} 1300&amp;nbsp;BC).&lt;ref&gt;Jane C. Waldbaum, ''From Bronze to Iron: The Transition from the Bronze Age to the Iron Age in the Eastern Mediterranean'' (Studies in Mediterranean Archaeology, vol. LIV, 1978).&lt;/ref&gt;

==21st Century BC==

In the Black Pyramid of Abusir, dating before 2000&amp;nbsp;BC, Gaston Maspero found some pieces of iron. In the funeral text of Pepi I, the metal is mentioned.&lt;ref name=&quot;EB1910CH&quot; /&gt; A sword bearing the name of pharaoh Merneptah as well as a battle axe with an iron blade and gold-decorated bronze shaft were both found in the excavation of Ugarit.&lt;ref&gt;Richard Cowen, ''The Age of Iron'' Chapter 5 in a series of essays on Geology, History, and People prepares for a course of the University of California at Davis. [http://mygeologypage.ucdavis.edu/cowen/~GEL115/115CH5.html Online version]&lt;/ref&gt; Tutankhamun's meteoric iron dagger with an iron blade was found in Tutankhamun's tomb, 13th century BC, was recently examined and found to be of meteoric origin.&lt;ref&gt;
{{cite journal
 | title =The meteoritic origin of Tutankhamun's iron dagger blade
 | journal =Meteoritics &amp; Planetary Science
 | volume =51
 | issue =7
 | pages =1301
 | publisher = Wiley Online
 | doi =10.1111/maps.12664| year =2016
 | last1 =Comelli
 | first1 =Daniela
 | last2 =d'Orazio
 | first2 =Massimo
 | last3 =Folco
 | first3 =Luigi
 | last4 =El-Halwagy
 | first4 =Mahmud
 | bibcode =2016M&amp;PS...51.1301C }}Free full text available.&lt;/ref&gt; The &quot;blade’s composition of iron, nickel and cobalt was an approximate match for a meteorite that landed in northern Egypt. The result “strongly suggests an extraterrestrial origin&quot;&quot;.&lt;ref name=xxwalshx&gt;{{cite journal
 | last =Walsh
 | first =Declan
 | title =King Tut’s Dagger Made of ‘Iron From the Sky,’ Researchers Say
 | journal =The New York Times
 | date =2 June 2016
 | url =https://www.nytimes.com/2016/06/03/world/middleeast/king-tuts-dagger-made-of-iron-from-the-sky-researchers-say.html
 | accessdate =4 June 2016}}&lt;/ref&gt;&lt;ref name=pankobe&gt;{{cite journal
 | last =Panko
 | first =Ben
 | title =King Tut’s dagger made from an ancient meteorite
 | journal =Science
 | date =2 June 2016
 | url =http://www.sciencemag.org/news/sifter/king-tut-s-dagger-made-ancient-meteorite
 | accessdate =5 June 2016}}&lt;/ref&gt;

&quot;In Berber, the name &quot;Siwa&quot; means &quot;prey bird and protector of sun god Amon-Ra.&quot; It is derived from the name of the indigenous inhabitants, Tiswan, who speak Tassiwit, a dialect related to Berber spoken in the Sahara and North Africa. Siwa is one of the most arid oases in western Egypt near the border of Libya at a depression of 18 meters below sea level, and it is 300 kilometers southwest of the Mediterranean port city of Marsa Matruh. The oasis is 82 kilometers long and has a width ranging between 2 and 20 kilometers. The oasis was occupied since Paleolithic and Neolithic times. It was first mentioned more than 2,500 years ago in the records of the pharaohs of the Middle and New Kingdoms (2050-1800 B.C. and 1570-1090 B.C.)&quot;&lt;ref name=Ilahiane&gt;{{ cite book
|author=Hsain Ilahiane
|title=Historical Dictionary of the Berbers (Imazighen)
|publisher=The Scarecrow Press
|location=Lanham, Maryland USA
|date=17 July 2006
|pages=360
|url=https://books.google.com/books?isbn=0810864908
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=0810864908
|accessdate=22 September 2017 }}&lt;/ref&gt;
{{clear}}

==23rd Century BC==

The earliest tentative evidence for iron-making is a small number of iron fragments with the appropriate amounts of carbon admixture found in the Proto-Hittite layers at Kaman-Kalehöyük and dated to 2200–2000&amp;nbsp;BC. Akanuma (2008) concludes that &quot;The combination of carbon dating, archaeological context, and archaeometallurgical examination indicates that it is likely that the use of ironware made of steel had already begun in the third millennium BC in Central Anatolia&quot;.&lt;ref&gt;{{cite journal
|last=Akanuma
|first=Hideo
|title=The Significance of Early Bronze Age Iron Objects from Kaman-Kalehöyük, Turkey
|journal=Anatolian Archaeological Studies
|volume=17
|pages=313–320
|year=2008
|url=http://www.jiaa-kaman.org/pdfs/aas_17/AAS_17_Akanuma_H_pp_313_320.pdf
}}&lt;/ref&gt;

==27th Century BC==
Egyptian lenses of the IV/V Dynasties, in Egypt, are eyes in funerary statues, dated to 2620-2400 BC.&lt;ref name=Enoch&gt;{{ cite journal
|author=Jay M. Enoch, Vasudevan Lakshminarayanan
|title=Duplication of unique optical effects of ancient Egyptian lenses from the IV/V Dynasties: lenses fabricated ca 2620–2400 BC or roughly 4600 years ago
|journal=Ophthalmic and Physiological Optics
|date=March 2000
|volume=20
|issue=2
|pages=126-30
|url=http://onlinelibrary.wiley.com/doi/10.1046/j.1475-1313.2000.00496.x/abstract
|arxiv=
|bibcode=
|doi=10.1046/j.1475-1313.2000.00496.x
|pmid=
|accessdate=2012-10-03 }}&lt;/ref&gt;
{{clear}}

==30th Century BC==

In the Mesopotamian states of Sumer, Akkad and Assyria, the initial use of iron reaches far back, to perhaps 3000 BC.&lt;ref name=&quot;EB1910CH&quot;&gt;Chisholm, H. (1910). The Encyclopædia Britannica. New York: The Encyclopædia Britannica Co.&lt;/ref&gt; One of the earliest smelted iron artifacts known was a dagger with an iron blade found in a Hattic tomb in Anatolia, dating from 2500&amp;nbsp;BC.&lt;ref name=cowen&gt;Richard Cowen () ''The Age of Iron'' Chapter 5 in a series of essays on Geology, History, and People prepares for a course of the University of California at Davis. [http://mygeologypage.ucdavis.edu/cowen/~GEL115/115CH5.html Online version].&lt;/ref&gt;

==31st Century BC==
[[Image:Pyramide von Hellinikon.jpg|thumb|right|300px|The Pyramid of Hellinikon is probably 5,000 b2k. Credit: [[:de:User:Schuppi|Schuppi]].]]
The pyramid on the right may be earlier than 5,000 b2k.
{{clear}}

==33rd Century BC==
Artemision lenses (plano-concave), Archaic levels, the Artemision of Ephesos, dated to 3300-1200 BC per D.G. Hogarth, ''Excavations at Ephesus: The Archaic Artemisia'' (London 1908) 210-11, pl. 46.

==32nd Century BC==
[[Image:Three iron beads from Gerzeh.jpg|right|thumb|250px|Photographs of three of the originally nine iron beads from Gerzeh, Lower Egypt, from left UC10738, UC10739 and UC10740. Credit: Thilo Rehren, Tamás Belgya, Albert Jambon, György Káli, Zsolt Kasztovszky, Zoltán Kis, Imre Kovács, Boglárka Maróti, Marcos Martinón-Torres, Gianluca Miniaci, Vincent C. Pigott, Miljana Radivojević, László Rosta, László Szentmiklósi, Zoltán Szőkefalvi-Nagy.]]
[[Image:Neutron radiography.jpg|left|thumb|250px|Comparison of neutron radiography and an optical photograph of an iron bead is shown. Credit: Thilo Rehren, Tamás Belgya, Albert Jambon, György Káli, Zsolt Kasztovszky, Zoltán Kis, Imre Kovács, Boglárka Maróti, Marcos Martinón-Torres, Gianluca Miniaci, Vincent C. Pigott, Miljana Radivojević, László Rosta, László Szentmiklósi, Zoltán Szőkefalvi-Nagy.]]

&quot;The earliest known iron artefacts are nine small beads securely dated to circa 3200 BC, from two burials in Gerzeh, northern Egypt.&quot;&lt;ref name=Rehren&gt;{{ cite journal
|author=Thilo Rehrena
|author2=Tamás Belgya
|author3=Albert Jambon
|author4=György Káli
|author5=Zsolt Kasztovszky
|author6=Zoltán Kis
|author7=Imre Kovács
|author8=Boglárka Maróti
|author9=Marcos Martinón-Torres
|author10=Gianluca Miniaci
|author11=Vincent C. Pigott
|author12=Miljana Radivojević
|author13=László Rosta
|author14=László Szentmiklósi
|author15=Zoltán Szőkefalvi-Nagy
|title=5,000 years old Egyptian iron beads made from hammered meteoritic iron
|journal=Journal of Archaeological Science
|date=December 2013
|volume=40
|issue=12
|pages=4785–92
|url=http://www.sciencedirect.com/science/article/pii/S0305440313002057
|arxiv=
|bibcode=
|doi=10.1016/j.jas.2013.06.002
|pmid=
|accessdate=2016-10-23 }}&lt;/ref&gt;

&quot;Since both tombs are securely dated to Naqada IIC–IIIA, c 3400–3100 BC (Adams, 1990: 25; Stevenson, 2009: 11–31), the beads predate the emergence of iron smelting by nearly 2000 years, and other known meteoritic iron artefacts by 500 years or more (Yalçın 1999), giving them an exceptional position in the history of metal use.&quot;&lt;ref name=Rehren/&gt;

The image on the left uses neutron radiography to show the metal underneath the corrosion.

&quot;Bead UC10738 [in the image on the right] has a maximum length of 1.5 cm and a maximum diameter of 1.3 cm, bead UC10739 is 1.7 cm by 0.7 cm, and bead UC10740 is 1.7 cm by 0.3 cm. All three beads are of rust-brown colour with a rough surface, indicative of heavy iron corrosion. Initial analysis by [proton–induced X–ray fluorescence] pXRF indicated an elevated nickel content of the surface of the beads, in the order of a few per cent, and their magnetic property suggested that iron metal may be present in their body (Jambon, 2010).&quot;&lt;ref name=Rehren/&gt;
{{clear}}

==36th Century BC==
&quot;c. 3500 B.C. Phoenicians cooking on sand discover glass.&quot;&lt;ref name=Brown&gt;{{ cite book
|author=Larry Brown
|title=An early history of the telescope. From 3500 B.C. until about 1900 A.D.
|publisher=Antique Telescopes
|location=
|date= 2005
|url=http://www.antiquetelescopes.org/history.html
|accessdate=2012-04-08 }}&lt;/ref&gt;

&quot;Natural glass has existed since the beginnings of time, formed when certain types of rocks melt as a result of high-temperature phenomena such as volcanic eruptions, lightning strikes or the impact of meteorites, and then cool and solidify rapidly. Stone-age man is believed to have used cutting tools made of obsidian (a natural glass of volcanic origin also known as hyalopsite, Iceland agate, or mountain mahogany) and tektites (naturally-formed glasses of extraterrestrial or other origin, also referred to as obsidianites).&quot;&lt;ref name=Infoserv&gt;{{ cite book
|author=Infoserv
|title=A Brief History of Glass
|publisher=Artenergy Publishing S.r.l.
|location=
|date=April 15, 2011
|url=https://web.archive.org/web/20110415194738/http://www.glassonline.com/infoserv/history.html
|accessdate=2014-06-06 }}&lt;/ref&gt;

&quot;According to the ancient-Roman historian Pliny (AD 23-79), Phoenician merchants transporting stone actually discovered glass (or rather became aware of its existence accidentally) in the region of Syria around 5000 BC [7,000 b2k]. Pliny tells how the merchants, after landing, rested cooking pots on blocks of nitrate placed by their fire. With the intense heat of the fire, the blocks eventually melted and mixed with the sand of the beach to form an opaque liquid.&quot;&lt;ref name=Infoserv/&gt;

&quot;The earliest man-made glass objects, mainly non-transparent glass beads, are thought to date back to around 3500 BC, with finds in Egypt and Eastern Mesopotamia. In the third millennium, in central Mesopotamia, the basic raw materials of glass were being used principally to produce glazes on pots and vases. The discovery may have been coincidental, with calciferous sand finding its way into an overheated kiln and combining with soda to form a coloured glaze on the ceramics. It was then, above all, Phoenician merchants and sailors who spread this new art along the coasts of the Mediterranean.&quot;&lt;ref name=Infoserv/&gt;

==40th Century BC==
[[Image:Prince Ra-Hotep and Princess Nefer-T (cropped).jpg|thumb|right|250px|The eyes of the princess have irises of rock-crystal enclosing a pupil. Credit: William Vaughn Tupper (1835 — 1898).]]
[[Image:All Gizah Pyramids.jpg|left|thumb|250px|This image shows the pyramids of [[w:Giza|Giza]]. Credit: [http://liberato.org Ricardo Liberato].]]
[[Image:Mysterious Giza... - panoramio.jpg|left|thumb|300px|This is a panorama of the pyramid complex at Gisa. Credit: [http://www.panoramio.com/user/2510042?with_photo_id=79145291 G Da].{{tlx|free media}}]]
[[Image:Monted'accoddisardegna.png|right|thumb|250px|Monte D' Accoddi is on Sardinia. Credit: Gianf84.]]
&quot;It is also only within recent years that the discovery of a seminal black kingdom in the Nile Valley, predating the Egyptian dynasties, has settled the question, once and for all, of the roots of classical Egyptian culture and technology (see ''Journal of African Civilizations'', vol. 4, no. 2, November 1982).&quot;&lt;ref name=Sertima&gt;{{ cite book
|author=Ivan Van Sertima
|title=The Lost Sciences of Africa: An Overview, In: ''Blacks in Science: Ancient and Modern''
|publisher=Transaction Publishers
|location=Pscataway, New Jersey, USA
|date= 2010
|editor=Ivan Van Sertima
|pages=5-26
|url=http://books.google.com/books?hl=en&amp;lr=&amp;id=RqOz55ZBfdkC&amp;oi=fnd&amp;pg=PA5&amp;ots=MJBIE2cQ8v&amp;sig=N3hYQoGoqp3ShdiSX1TZppxe05E
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=978-0-87855-941-1
|accessdate=2012-10-04 }}&lt;/ref&gt;

“The Great Pyramid stands on the northern edge of the Giza Plateau, [60.4 m] 198 feet above sea level”.&lt;ref name=Nikolic&gt;{{ cite book
|author=Petko Vidusa Nikolic, Petko Nikolic Vidusa
|title=The Great Pyramid and the Bible : Earth's Measurements
|publisher=Mystik Book
|location=Kitchener, Canada
|date= 2005
|editor=
|pages=65
|url=
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=0973237147
}}&lt;/ref&gt;

A step pyramid exists in the archaeological site of Monte d'Accoddi, in Sardinia, dating to the 4th millennium BC, shown in the image on the right: &quot;a trapezoidal platform on an artificial mound, reached by a sloped causeway. At one time a rectangular structure sat atop the platform ... the platform dates from the Copper Age (c. 2700–2000 BC), with some minor subsequent activity in the Early Bronze Age (c. 2000–1600 BC). Near the mound are several standing stones, and a large limestone slab, now at the foot of the mound, may have served as an altar.&quot;&lt;ref&gt;{{cite book
|last=Blake
|first=Emma
|title=The archaeology of Mediterranean prehistory
|date=2004
|publisher=Wiley Blackwell
|isbn=978-0-631-23268-1
|url=https://books.google.com/?id=F15vfrJq8LUC&amp;pg=PA117&amp;dq=Monte+d%27Accoddi#v=onepage&amp;q=Monte%20d%27Accoddi&amp;f=false
|author2=Arthur Bernard Knapp
|accessdate=31 August 2011
|page=117
}}&lt;/ref&gt;
{{clear}}

==41st Century BC==
[[Image:Menhirmonted'accoddi.png|right|thumb|300px|View is from the base of Monte d'Accoddi, Sardinia. Credit: Gianf84.{{tlx|free media}}]]
[[Image:Sassari - Complesso prenuragico di Monte d'Accoddi (04).JPG|left|thumb|300px|This raised platform may have been for worship of the Sun god. Credit: [[commons:User:Discanto|Gianni Careddu]].{{tlx|free media}}]]
[[Image:Ice-core-isotope.png|thumb|left|300px|Last glacial period is seen in ice core data from Antarctica (top) and Greenland (bottom). Credit: [[w:user:Leland McInnes|Leland McInnes]].]]
[[Image:Mediterranean Bathymetric map.png|center|thumb|500px|This bathymetric map of the Mediterranean Sea shows the likely surface above sea level during the last ice age in orange-brown. Credit: Jean Mascle &amp; Georges Mascle, Geological and Morpho-Tectonic Map of the Mediterranean Domain, Digital Terrain Model (DTM), at 500m, DTM at 1500m GEBCO Atlas (General Bathymetric Chart of the Oceans), Morpho-bathymetric synthesis of the Mediterranean Sea, CIESM and IFREMER.{{tlx|fairuse}}]]
The original structure shown in left and right images may have been built by the Ozieri culture or earlier ''c.'' 4,000-3,650 BC.

Paleolithic workshops on Sardinia suggest a hominin presence in Sardinia between 450,000 and 10,000 years ago.

During the last ice age sea levels were likely lower than 200 meters, at that time Sardinia and Corsica formed a single large island, separated from Tuscany by a narrow arm of sea, marked in yellow between Sardinia and Tusscany, perhaps below 1,000 m in depth during the last ice age.

Comparison of temperature proxies for ice cores from Antarctica and Greenland for 140,000 years are shown on the left. Greenland ice cores use delta &lt;sup&gt;18&lt;/sup&gt;O, while Antarctic ice cores use delta &lt;sup&gt;2&lt;/sup&gt;H. Note that the Dansgaard-Oeschger events in the Greenland ice core between 20,000 and 110,000 years ago barely register (if at all) in the corresponding Antarctic record.
{{clear}}

==45th Century BC==

&quot;In the Chalcolithic/earliest Bronze Age I period (c. 4500±3000 cal BC), copper was mined in open galleries from the massive brown sandstone deposit, which consisted of thick layers of the copper carbonate malachite and chalcocite, a copper sulphide.&quot;&lt;ref name=Ottaway&gt;{{ cite journal
|author=B.S. Ottaway
|title=Innovation, production and specialization in early prehistoric copper metallurgy
|journal=European Journal of Archaeology
|year=2001
|volume=4
|issue=1
|pages=87-112
|url=http://www.tandfonline.com/doi/abs/10.1179/eja.2001.4.1.87
|arxiv=
|bibcode=
|doi=10.1179/eja.2001.4.1.87
|pmid=
|accessdate=2016-10-23 }}&lt;/ref&gt;
{{clear}}

==50th Century BC==

&quot;An archaeological site in [Serbia] has shown its metal. This ancient settlement contains the oldest securely dated evidence of copper making, from 7,000 years ago, and suggests that copper smelting may been invented in separate parts of Asia and Europe at that time rather than spreading from a single source.&quot;&lt;ref name=Bower&gt;{{ cite journal
|title=Serbian site may have hosted first copper makers
|author=Bruce Bower
|month=17 July
|year=2010
|journal=ScienceNews
|url=http://www.sciencenews.org/view/generic/id/60563/description/Serbian_site_may_have_hosted_first_copper_makers
|accessdate=22 April 2017}}&lt;/ref&gt;

==55th Century BC==

&quot;Until now, experts said that only stone was used in the Stone Age and that the Copper Age came a bit later. Our finds, however, confirm that metal was used some 500 to 800 years earlier, [which indicates that humans were using metals in Europe by 7,500 years ago].&quot;&lt;ref name=Cvetkovic&gt;{{ cite book
|url=http://www.thaindian.com/newsportal/india-news/ancient-axe-find-suggests-copper-age-began-earlier-than-believed_100105122.html
|author=Julka Kuzmanovic-Cvetkovic
|title=Ancient axe find suggests Copper Age began earlier than believed
|date=8 October 2008
|accessdate=2017-10-13 }} &lt;/ref&gt;

==Hypotheses==
{{main|Hypotheses}}
# The sciences do not need to follow the humanities, religion, or the arts to be beneficial to humanity.
# The scientific method is not a recent occurrence.

==See also==
{{div col|colwidth=12em}}
* [[Dominant group/Atmospheric sciences]]
* [[What is science?]]
{{Div col end}}

==References==
{{reflist|2}}

==External links==
* [http://www.bing.com/search?q=&amp;go=&amp;qs=n&amp;sk=&amp;sc=8-15&amp;qb=1&amp;FORM=AXRE Bing Advanced search]
* [http://books.google.com/ Google Books]
* [http://scholar.google.com/advanced_scholar_search?hl=en&amp;lr= Google scholar Advanced Scholar Search]
* [http://www.jstor.org/ JSTOR]
* [http://www.lycos.com/ Lycos search]
* [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center]
* [http://www.ncbi.nlm.nih.gov/sites/gquery NCBI All Databases Search]
* [http://www.osti.gov/ Office of Scientific &amp; Technical Information]
* [http://psycnet.apa.org/ PsycNET]
* [http://www.ncbi.nlm.nih.gov/pccompound PubChem Public Chemical Database]
* [http://www.questia.com/ Questia - The Online Library of Books and Journals]
* [http://online.sagepub.com/ SAGE journals online]
* [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System]
* [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search]
* [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database]
* [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate]
* [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA]
* [http://www.springerlink.com/ SpringerLink]
* [http://www.tandfonline.com/ Taylor &amp; Francis Online]
* [http://www.wikidoc.org/index.php/Main_Page WikiDoc The Living Textbook of Medicine]
* [http://onlinelibrary.wiley.com/advanced/search Wiley Online Library Advanced Search]
* [http://search.yahoo.com/web/advanced Yahoo Advanced Web Search]

&lt;!-- footer templates --&gt;
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[[Category:History of Sciences/Lectures]]
[[Category:Sciences/Lectures]]
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      <text bytes="5791" sha1="k3ip0pqlo66d5sdkd1bm4kfiv1awk2q" xml:space="preserve">{{title|General feedback about&lt;br&gt;book chapters}}
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* General feedback about the [[Motivation and emotion/Book/2026|2026 student-authored book chapters]]
* Detailed feedback about each individual chapter is available on the respective talk pages
{{RoundBoxBottom}}

__TOC__

==Overall==
# The overall quality of chapters was reasonably good, but there was a notably wide range in quality from poor to outstanding
# The most common issues were:
## Not directly addressing the topic (as specified in the title and sub-title)
## Making unacknowledged overuse use of [[Motivation and emotion/Assessment/Using generative AI|genAI]], rather than prioritising human writing based on reading of original sources
## Insufficient review and synthesis of the best research about the topic (weaker chapters were commonly theory-heavy and research-light)
## No or minimal engagement in making social contributions with direct links to evidence

==Overview==
# The best overviews consisted of:
## A scenario/case study which appealed to a broad audience in a feature box with a relevant image
## A straighforward, compelling description of the psychological problem or phenomenon
## Clear and relevant focus questions that closely aligned with the top-level headings.

==Theory==
# Most chapters used a well-selected range of relevant theories
# Theories were generally well explained, but stronger chapters exhibit more incisive theoretical depth based on reading and understanding of primary sources
# Almost chapters could be improved by more strongly embedding links to related chapters
# The strongest chapters illustrated theoretical princples through practical examples

==Research==
# Depth of critical review of research was the single most distinguishing characteristic of the strongest chapters
# Most chapters reviewed some relevant research, but often the review of research could have been more relevant and/or indepth
# More emphasis should have been placed on major reviews such as meta-analyses
# To improve critical review of research, for key studies, report details about the sample, country, method, and strength of observed effects
# Weaker chapters often lacked sufficient citation of academic peer-reviewed sources to support claims

==Integration==
# The best chapters had excellent integration between theory and research
# Weakers chapters tended to rely more heavily on theory than research, or covered research as a stand-alone topic rather than integrated review that synthesised theory and research

==Conclusion==
# Often the weaker conclusions read like generic genAI summaries
# The best chapters addressed the focus questions and provided explicit take-home messages

==Style==
===Written expression===
# The overall quality of written expression was reasonably good, but varied widely
# The most common problems were:
## grammar and spelling
### poor grammar—there's many ways to get help to develop professional-level written expression (e.g., Studiosity, genAI)
### use 3rd person, rather than 1st person, perspective[https://www.grammarly.com/blog/first-second-and-third-person/]
### use past tense when describing research studies, although implications of findings could be in the present tense
### use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should] (they are part of APA style)
### use [https://grammar.yourdictionary.com/punctuation/apostrophe-rules.html ownership (possessive) apostrophes]
### Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling].
## insufficient citation—psychological science community requires claims to be supported by cited evidence and sources
# The most common APA style reference formatting areas for improvement were:
### capitalisation
### italicisation

===Learning features===
# The use of learning features varied widely between chapters
# The best chapters were interconnected with other Wikimedia resources by using:
## embedded interwiki links to Wikipedia articles help to connect reader to further information about key concepts
## embedded interwiki links to related Wikiversity [[Motivation and emotion/Book|book chapters]]
# Figures were reasonably well used, with many students uploading and contributing their own images (thankyou!)
# Tables were less commonly used, but were strongest when thoughtfully constructed and weaker when constructed by genAI and lacking in acknowledgement of sources
# Additional scenarios/case studies in feature boxes were underused to provide illustrative examples
# The best chapters used distributed quiz questions about take-home messages rather than being presented at the end

==Social contributions==
# The amount and quality of contributions varied widely: The use of social contributions distinguished sharply between genuinely and casually engaged authors—some students made no contributions whilst some excelled and were awarded bonus marks 
# Many valuable improvements were made to past and current chapters
# Unless there were ''direct links'' to evidence, no marks were provided

== See also ==
* [[Motivation and emotion/Assessment/Chapter/Feedback/2021|2021 feedback]]
* [[Motivation and emotion/Assessment/Chapter/Feedback/2022|2022 feedback]]
* [[Motivation and emotion/Assessment/Chapter/Feedback/2023|2023 feedback]]
* [[Motivation and emotion/Assessment/Chapter/Feedback/2024|2024 feedback]]
* [[Motivation and emotion/Assessment/Chapter/Feedback/2025|2025 feedback]]
* [[Motivation and emotion/Assessment/Chapter|Book chapter - Guidelines]]&lt;!--
* [[Motivation and emotion/Assessment/Multimedia/Feedback|Multimedia presentation feedback]] --&gt;

[[Category:Motivation and emotion/Assessment/Chapter]]</text>
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  <page>
    <title>Web Browser</title>
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      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-59650-28|~2026-59650-28]] ([[User_talk:~2026-59650-28|talk]]) to last version by [[User:CommonsDelinker|CommonsDelinker]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="650" sha1="28bwvy57bpmlfys6bzh3jweu3bgbxza" xml:space="preserve">[[File:Google Chrome.svg|right|100px|Chrome logo]]
[[File:Firefox logo, 2019.svg|right|100px|Firefox logo]]
[[File:Microsoft Edge logo (2019).svg|right|100px|Edge logo]]
[[File:Opera 2015 icon.svg|right|100px|Opera logo]]

A web browser (commonly referred to as a browser) is a software application for retrieving, presenting, and traversing information resources on the World Wide Web.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;

== Lessons ==
* [[Computer Skills/Basic/Internet]]
* [[Internet Fundamentals/Web Browsers]]
* [[IC3/Internet Fundamentals]]

== See Also ==
* [[Wikipedia: Web browser]]

== References ==
{{Reflist}}

[[Category:Web browsers]]</text>
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    <title>Talk:Universal Bibliography</title>
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      <comment>/* */ Add</comment>
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      <text bytes="10885" sha1="orjs5n27et085dljumawwteu7yqdw49" xml:space="preserve">Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC)

For future use:
*Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums
*Reprography, lithography, engraving
*Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology
*Languages, philology, ancient languages, medieval languages, dead languages, extinct languages
*Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B
*Indo-European languages; Italo-Celtic; Celtic languages
*Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite
*Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries
*Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology
*Tide tables

*George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ]
*[[s:Page:Cornwall (Mitton).djvu/230|&quot;Some Books on Cornwall&quot;]]
*[[s:Author:Richard Colt Hoare]]
*Yorkshire literature

*Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&amp;pg=PA1#v=onepage&amp;q&amp;f=false] (regional catalogues of incunabula in the public libraries of France)
* [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC)

For future use:

*Palmer's Index to The Times
**editions:2Faf22c-dSwC

*Routledge Handbook of Soviet and Russian Military Studies

France:
*Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries)

Creuse:
*Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ]
*Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C]
*Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Journal du département de la Creuse (1807-1819)

Bibliography:
*[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell &amp; Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA525#v=onepage&amp;q&amp;f=false]

*Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC]

Series:
*Studies in Bibliography. Formerly &quot;Papers in Bibliography&quot;. (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: &quot;A History of Studies in Bibliography: The First Fifty Volumes&quot; in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ]
**Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA526#v=onepage&amp;q&amp;f=false]

Library associations:
*IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&amp;pg=PP4#v=onepage&amp;q&amp;f=false]
*Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Serials:
*Serials Review

Nostgia
*Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html]
*Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ].
*1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&amp;pg=PA25#v=onepage&amp;q&amp;f=false].
*[https://books.google.co.uk/books?id=COhVAAAAIBAJ&amp;pg=PA20#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&amp;pg=PA3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&amp;pg=PA31#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&amp;pg=PA27&amp;article_id=4890,1282350#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&amp;pg=PA29#v=onepage&amp;q&amp;f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&amp;pg=PA27#v=onepage&amp;q&amp;f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&amp;pg=PA7#v=onepage&amp;q&amp;f=false] 
*1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&amp;pg=PA8#v=onepage&amp;q&amp;f=false]
*1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia]
*[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC.
*[https://www.rd.com/article/tv-shows-like-stranger-things/ Stranger Things Is Over—Now What?! Here Are 7 Shows to Watch to Fill the Void]. Reader's Digest. 1 January 2026.
*Anne Latreille.
[https://trove.nla.gov.au/newspaper/article/47808279 &quot;There's a Boom in Nostalgia&quot;]. [[w:The Australian Women's Weekly|The Australian Women's Weekly]]. 29 March 1972. p 7.
*The Past in the Present: Proceedings of the 5th Annual British and Cultural Studies Conference, Oldenburg 1994

Other
*[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943)
**Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々)
**Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅)

Search terms:
*&quot;press in japan&quot;
*&quot;japanese press&quot;
*&quot;press of japan&quot;
*&quot;journalism of japan&quot;
*&quot;journalism in japan&quot;
*&quot;journalists in japan&quot;
*&quot;japanese journalism&quot;

Generations
*[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018.

Planet Earth
*[[w:Encyclopedia of Earth|Encyclopedia of Earth]]

Crime
*The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ]

Showa era
*懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ]
*少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ]

Fading of the Showa era
*[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025.
*[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026.

Showa retro packaging
*[[w:ja:初見健一|Kenichi Hatsumi]]. 昭和のレトロパッケージ. [[w:ja:グラフィック社|Graphic-sha]]. 2016. ISBN 9784766128741. [https://books.google.co.uk/books?id=Ch5wjwEACAAJ]

Shotengai
*[https://www.theguardian.com/world/2025/jul/18/cult-of-convenience-how-tokyos-retro-shotengai-arcades-are-falling-victim-to-gentrification ‘Cult of convenience’: how Tokyo’s retro shotengai arcades are falling victim to gentrification]. The Guardian. 18 July 2025.

Shakespeare
*Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885.
[https://books.google.co.uk/books?id=GSZLAQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false].
*Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false]

*Best films:
[https://www.empireonline.com/movies/features/best-movies-of-all-time-us/]

Survivability of past atmosphere
*[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022.
*[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. 
*[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014.

Philology
*W Pökel. Philologisches Schriftsteller-Lexikon. 1882.  [https://books.google.co.uk/books?id=2jAaAAAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Rivista di filologia e di istruzione classica [https://books.google.co.uk/books?id=M6AzAQAAMAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

Literature, science and art
*[[The Academy (periodical)|The Academy]] [https://books.google.co.uk/books?id=rIQWRoLtj3oC&amp;pg=PA5#v=onepage&amp;q&amp;f=false]

[[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC)</text>
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      <text bytes="10887" sha1="0eyq0fjakrir78cmzmdrt9ccams6f04" xml:space="preserve">Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC)

For future use:
*Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums
*Reprography, lithography, engraving
*Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology
*Languages, philology, ancient languages, medieval languages, dead languages, extinct languages
*Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B
*Indo-European languages; Italo-Celtic; Celtic languages
*Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite
*Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries
*Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology
*Tide tables

*George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ]
*[[s:Page:Cornwall (Mitton).djvu/230|&quot;Some Books on Cornwall&quot;]]
*[[s:Author:Richard Colt Hoare]]
*Yorkshire literature

*Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&amp;pg=PA1#v=onepage&amp;q&amp;f=false] (regional catalogues of incunabula in the public libraries of France)
* [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC)

For future use:

*Palmer's Index to The Times
**editions:2Faf22c-dSwC

*Routledge Handbook of Soviet and Russian Military Studies

France:
*Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries)

Creuse:
*Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ]
*Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C]
*Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Journal du département de la Creuse (1807-1819)

Bibliography:
*[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell &amp; Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA525#v=onepage&amp;q&amp;f=false]

*Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC]

Series:
*Studies in Bibliography. Formerly &quot;Papers in Bibliography&quot;. (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: &quot;A History of Studies in Bibliography: The First Fifty Volumes&quot; in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ]
**Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA526#v=onepage&amp;q&amp;f=false]

Library associations:
*IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&amp;pg=PP4#v=onepage&amp;q&amp;f=false]
*Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Serials:
*Serials Review

Nostgia
*Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html]
*Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ].
*1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&amp;pg=PA25#v=onepage&amp;q&amp;f=false].
*[https://books.google.co.uk/books?id=COhVAAAAIBAJ&amp;pg=PA20#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&amp;pg=PA3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&amp;pg=PA31#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&amp;pg=PA27&amp;article_id=4890,1282350#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&amp;pg=PA29#v=onepage&amp;q&amp;f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&amp;pg=PA27#v=onepage&amp;q&amp;f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&amp;pg=PA7#v=onepage&amp;q&amp;f=false] 
*1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&amp;pg=PA8#v=onepage&amp;q&amp;f=false]
*1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia]
*[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC.
*[https://www.rd.com/article/tv-shows-like-stranger-things/ Stranger Things Is Over—Now What?! Here Are 7 Shows to Watch to Fill the Void]. Reader's Digest. 1 January 2026.
*Anne Latreille.
[https://trove.nla.gov.au/newspaper/article/47808279 &quot;There's a Boom in Nostalgia&quot;]. [[w:The Australian Women's Weekly|The Australian Women's Weekly]]. 29 March 1972. p 7.
*The Past in the Present: Proceedings of the 5th Annual British and Cultural Studies Conference, Oldenburg 1994

Other
*[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943)
**Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々)
**Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅)

Search terms:
*&quot;press in japan&quot;
*&quot;japanese press&quot;
*&quot;press of japan&quot;
*&quot;journalism of japan&quot;
*&quot;journalism in japan&quot;
*&quot;journalists in japan&quot;
*&quot;japanese journalism&quot;

Generations
*[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018.

Planet Earth
*[[w:Encyclopedia of Earth|Encyclopedia of Earth]]

Crime
*The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ]

Showa era
*懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ]
*少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ]

Fading of the Showa era
*[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025.
*[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026.

Showa retro packaging
*[[w:ja:初見健一|Kenichi Hatsumi]]. 昭和のレトロパッケージ. [[w:ja:グラフィック社|Graphic-sha]]. 2016. ISBN 9784766128741. [https://books.google.co.uk/books?id=Ch5wjwEACAAJ]

Shotengai
*[https://www.theguardian.com/world/2025/jul/18/cult-of-convenience-how-tokyos-retro-shotengai-arcades-are-falling-victim-to-gentrification ‘Cult of convenience’: how Tokyo’s retro shotengai arcades are falling victim to gentrification]. The Guardian. 18 July 2025.

Shakespeare
*Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885.
[https://books.google.co.uk/books?id=GSZLAQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false].
*Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false]

*Best films:
[https://www.empireonline.com/movies/features/best-movies-of-all-time-us/]

Survivability of past atmosphere
*[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022.
*[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. 
*[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014.

Philology
*W Pökel. Philologisches Schriftsteller-Lexikon. 1882.  [https://books.google.co.uk/books?id=2jAaAAAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Rivista di filologia e di istruzione classica [https://books.google.co.uk/books?id=M6AzAQAAMAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

Literature, science and art
*[[w:The Academy (periodical)|The Academy]] [https://books.google.co.uk/books?id=rIQWRoLtj3oC&amp;pg=PA5#v=onepage&amp;q&amp;f=false]

[[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC)</text>
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    <title>Cyberculture/Social Media/TikTok</title>
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      <comment>Add educational information about TikTok repost management and third-party tools</comment>
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      <text bytes="2501" sha1="kt5deu6rqvhms6r50owrogit8o8ncl3" xml:space="preserve">TIKTOK

Overview - 

TikTok is an iOS and Android social media video app for creating and sharing short lip-sync, comedy, and talent videos. The application allows users to create short music and lip-sync videos of 3 to 15 seconds and short looping videos of 3 to 60 seconds. &lt;nowiki/&gt;&lt;ref&gt;{{Cite web|url=https://theindustryobserver.thebrag.com/tik-tok-most-popular-app/|title=TikTok is fast becoming the most popular app in the world|date=2018-11-04|website=The Industry Observer|language=en-AU|access-date=2019-09-12}}&lt;/ref&gt;
It was launched in September 2016 by ByteDance under the name of ‘Douyin’ for Chinese markets. In September 2017, it was relaunched under the name ‘TikTok’ to cater to demand in foreign markets.
On 9th November 2017, ByteDance invested upto $1billion to acquire Music.ly, a Shanghai based start up with an
&lt;references group=&quot;https://theindustryobserver.thebrag.com/tik-tok-most-popular-app&quot; /&gt;office in Santa Monica, California. 

Owing to it’s popularity as a social media platform that targets the US teenage market, the acquisition helped TikTok break into the west.


== Third-party TikTok Repost Management Tools ==

Third-party tools can provide additional ways for users to manage their repost history on social media platforms. These tools may offer functions such as reviewing reposted content, automating repetitive management tasks, and monitoring the progress of content removal.

One example is ClearTok, an independent third-party tool designed to help TikTok users manage and remove their reposted videos. It is available through browser extensions and mobile applications. Its features include bulk repost removal, progress monitoring, and exporting records of removed video URLs.

ClearTok's official website provides further information about its functionality, supported platforms, privacy practices, and technical approach:

[https://cleartok.io/ ClearTok – TikTok Repost Remover]

ClearTok is an independent commercial product and is not affiliated with or endorsed by TikTok or ByteDance.

When evaluating third-party tools, users should consider account security, privacy, compatibility, and the limitations of automated interactions with social media platforms.


REFERENCES AND SOURCES
&lt;nowiki/&gt;&lt;ref&gt;{{Cite web|url=https://skparadize.in/how-to-earn-money-by-tik-tok/|title=How to earn money from tik tok|date=2020-03-04|website=Skparadize|language=en-AU|access-date=2020-03-04}}&lt;/ref&gt;
&lt;references /&gt;

[[Category:Social media]]</text>
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      <text bytes="2702" sha1="nfacjcqnj3zu8dt4vl5m43yrztyb71m" xml:space="preserve">TIKTOK

Overview - 

TikTok is an iOS and Android social media video app for creating and sharing short lip-sync, comedy, and talent videos. The application allows users to create short music and lip-sync videos of 3 to 15 seconds and short looping videos of 3 to 60 seconds. &lt;nowiki/&gt;&lt;ref&gt;{{Cite web|url=https://theindustryobserver.thebrag.com/tik-tok-most-popular-app/|title=TikTok is fast becoming the most popular app in the world|date=2018-11-04|website=The Industry Observer|language=en-AU|access-date=2019-09-12}}&lt;/ref&gt;
It was launched in September 2016 by ByteDance under the name of ‘Douyin’ for Chinese markets. In September 2017, it was relaunched under the name ‘TikTok’ to cater to demand in foreign markets.
On 9th November 2017, ByteDance invested upto $1billion to acquire Music.ly, a Shanghai based start up with an
&lt;references group=&quot;https://theindustryobserver.thebrag.com/tik-tok-most-popular-app&quot; /&gt;office in Santa Monica, California. 

Owing to it’s popularity as a social media platform that targets the US teenage market, the acquisition helped TikTok break into the west.


== Third-party TikTok Repost Management Tools ==

Third-party tools can provide additional ways for users to manage their repost history on social media platforms. These tools may offer functions such as reviewing reposted content, automating repetitive management tasks, and monitoring the progress of content removal.

One example is ClearTok, an independent third-party tool designed to help TikTok users manage and remove their reposted videos. It is available through browser extensions and mobile applications. Its features include bulk repost removal, progress monitoring, and exporting records of removed video URLs.

ClearTok's official website provides further information about its functionality, supported platforms, privacy practices, and technical approach:

[https://cleartok.io/ ClearTok – TikTok Repost Remover]

ClearTok is an independent commercial product and is not affiliated with or endorsed by TikTok or ByteDance.

When evaluating third-party tools, users should consider account security, privacy, compatibility, and the limitations of automated interactions with social media platforms.


REFERENCES AND SOURCES
&lt;nowiki/&gt;&lt;ref&gt;{{Cite web|url=https://skparadize.in/how-to-earn-money-by-tik-tok/|title=How to earn money from tik tok|date=2020-03-04|website=Skparadize|language=en-AU|access-date=2020-03-04}}&lt;/ref&gt;
&lt;ref&gt;{{Cite web|url=https://skparadize.in/how-to-earn-money-by-tik-tok/|title=How to Delete All Reposts on TikTok at Once|date=2025-07-01|website=ClearTok|language=en-AU|access-date=2025-07-01}}&lt;/ref&gt;
&lt;references /&gt;

[[Category:Social media]]</text>
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  <page>
    <title>Social Victorians/People/Gwladys Robinson</title>
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      <text bytes="156412" sha1="aut0b72li70aibv4pxp1sgmvpf9sngo" xml:space="preserve">{{Short description|Dress worn by Queen Victoria at her wedding to Prince Albert in 1840}}
= Sandbox =
Page to draft revisions for Wikipedia articles.

For Gwladys Robinson, see Gwladys Lowther Robinson, [[Social Victorians/People/Ripon|Marchioness of Ripon]] and, earlier, [[Social Victorians/People/Lowther|Countess of Lonsdale]]
==References==
{{reflist|2}}

[[Category:1840 works]]
[[Category:Royal wedding dresses|Victoria Queen]]
[[Category:1840s fashion]]
[[Category:British royal attire]]
[[Category:Dresses in the Royal Collection of the United Kingdom|Victoria, Wedding]]
[[Category:Diamond Jubilee of Queen Victoria]]

= Victorian fashion =

== Notes and Quotes ==

* Women followed Paris fashion in Europe and America, but London tailors led the way for men’s fashion. The Industrial Revolution actually impacted men’s fashion more than women’s.
* Sewing machines in factories began mass producing cheap clothing for the largest group in the population, working class men.
* Ready-made clothing affected the social structure, the economy, and gender differentiation.
* Paper patterns, introduced in fashion magazines that targeted women, along with domestic sewing machines had more impact on women’s fashion than men’s. Haute couture was still bespoke (being individually constructed by tailors or dressmakers or fashion houses like Maison Worth).

Sewing Machines and Clothes

Before the industrial revolution wealthy people paid others to sew their clothes by hand and those who were of much lower economic status did all the manual labor necessary to feed and clothe the rich minority. The working class had to make all their own clothes. Sewing, however, came at the end of the process.  They began with raw wool cut from the backs of sheep or gathering flax. Only the wealthy could afford to buy fabric. They followed processes for cleaning and spinning the wool into thread and the processes of soaking and pulling out the fibrous strips to make into thread for linen fabric. After they had thread, it was tied to a loom so it could be woven into cloth. Then it had to be cut or torn into pieces. For many families scissors were a luxury. And the last step is sewing the pieces together to create a garment.

Machines could not only sew, they could process the raw materials into thread and weave it into fabric much  quicker. Regarding clothing, working class men and their families benefitted greatly from ready-made clothes. Manufacturers marketed to them because they were the largest class. For example,&lt;blockquote&gt;–Men’s shirts, made by machine, take 3 hours and 10 minutes each to do; by hand, 15 hours and 20 minutes; a lady’s dress, by machine, takes 12 hours 4 minutes; by hand 20 hours 35 minutes; . . . a common frock-coat, by the machine occupies 17 hours 20 minutes; by the hand, 27 hours and 40 minutes: a linen vest, by machine, 3 hours 44 minutes; by hand 7 hours 18 minutes; a fine overcoat, by machine, 28 hours 13 minutes; by hand 31 hours 20 minutes; . . . &lt;ref&gt;Mitchell, Rebecca, ed. &quot;22. 'Progress of the Sewing Machine,' ''Bow Bells'' (19 April 1865), p. 283.&quot; Section 6: Production and Industry. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.&lt;/ref&gt;&lt;/blockquote&gt;

=== Industrial Revolution and Technological Advancements ===

* the railroad
* mass production of fabrics for working-class men's clothing
** Jacquard looms
* sewing machines
* aniline dyes, including black

=== Expanding Middle Class ===

* gender roles getting more clearly defined and rigid (referring back to page overview)
* &quot;There was, however, by the middle of the century a prosperous rising middle class who created a new demand for gentlemanly well-cut comfortable clothes-businessmen who had no time for the foibles of fashion-and so the earlier dashing lines sobered up, tailors co-operated by concentrating on fit and execution, and by the end of the century men's clothes had become practical, dull and uniform. Superb materials and expert tailoring alone distinguished a gentleman of fashion from his fellows.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1577 of 2116)
*
=== &quot;Outward appearance reflected inner qualities&quot; ===
Brent Shannon:&lt;blockquote&gt;&quot;Costume,&quot; wrote Max Beerbohm in 1896, &quot;enables us to classify any 'professional man' at a glance, be he lawyer, leech or who not&quot; (24–25). A man's profession and class were read by his jacket, his hat, what he rode in, and how he carried himself. &quot;Perhaps there is a tendency among Englishmen to judge a man too much by the shape of his hat or the kind of collar he wears,&quot; conduct author John Wanamaker confessed; &quot;But one must remember that in England if you ''wear'' the wrong thing, you will probably ''do'' the wrong thing, and generally ''be'' the wrong thing&quot; (1).&lt;sup&gt;11&lt;/sup&gt;

Such assertions were predicated on the powerful Victorian conviction that outward appearance reflected inner qualities.&lt;ref&gt;{{Cite book|title=The Cut of His Coat: Men, Dress, and Consumer Culture in Britain, 1860–1914|last=Shannon|first=Brent Alan|publisher=Ohio University Press|year=2006|location=Athens, Ohio}}&lt;/ref&gt; (148)&lt;/blockquote&gt;

==Women's fashion==

== Men's fashion ==
(Two intros: one to Victorian Fashion, giving background for both women's and men's fashions; two to the decades for men's fashion)

Among the influences on men's styles:
* differentiation of genders (domestic ideology, ideology of the spheres)
* the &quot;Great Male Renunciation,&quot; the French Revolution and English tailoring
* industrial revolution: mass production of fabrics and articles of clothing, especially for working-class men
* sports and country clothing
* Albert Edward, Prince of Wales, later King Edward VII, influence went beyond England; tailors led by him led fashion for the upper classes
* then later, cult of youth
* military uniforms
* comfort and convenience
* evolving definitions of gender, both femininity and masculinity evolved as concepts. The changing definitions of masculinity affected men's clothing. For the growing and rising middle classes, gender roles became more and more rigid around the concept of the separate spheres, the public for men and the private sphere for women.
Cultural sources for men's clothing and men's fashion: industrial revolution; expanding middle class; technological advancements; an association in the culture between outward appearance and inward nature; French Revolution; Albert Edward, Prince of Wales

Citation for Victorian Hell: &lt;ref name=&quot;:29&quot;&gt;{{Cite web|url=https://victorianhell.substack.com/p/victorian-mens-fashion-history-and|title=Victorian Men’s Fashion History and Clothing Guide|last=&quot;It's Monty, Actually&quot;|date=31 October 2025|website=Victorian Hell|access-date=7 September 2026}}&lt;/ref&gt;

=== Waistcoats ===

*c. 1830, dark brown or black w faces, gold buttons, standing collar: File:Waistcoat MET 38.68 F.jpg
* c. 1830, beige and white woven fabric with rose buds stamped or embroidered??: File:Waistcoat MET 19.181.4 F.jpg
* 1830–1835, dotted and floral stamped fabric, brown and tan: File:Waistcoat MET 1976.254.1.jpg
*
* c. 1840, flowered fabric: File:Waistcoat MET 1979.346.45 F.jpg
* c. 1840, red, jacquard-like, double-breasted: File:Waistcoat MET 1979.346.47 F.jpg

* 1840s, white brocade: File:Waistcoat MET 1980.132.8 F.jpg
* 1840s, dark blue, red, green? stamped?: File:Waistcoat MET 1976.254.1.jpg
* Mid 1840s, brown, embroidered: File:Waistcoat MET 24.160.3 F.jpg
* late 1840s, embossed white with hash thingies: File:Waistcoat MET 1978.88.9 F.jpg
* 1845 waistcoat made from blue Chinese silk damask: File:1845 waistcoat made from blue Chinese silk damask.jpg
* 1845–1855, white brocade wedding waistcoat, piping on the edges: File:Waistcoat MET 1986.373.3 F.jpg
* '''1849–1860''', black w gold lining, ribbon: File:Waistcoat MET 1978.477.25 F.jpg
*
* 1850, print of St. James's Park: File:Print (BM 1880,1113.2351).jpg
* 1850s, white and blue, w blue floral design stamped on white-and-grey stamped on white fabric: File:Waistcoat MET 1978.477.28 F.jpg
*c. 1853, brown and tan plaid, with red squares: File:Waistcoat MET 1984.90.2 F.jpg
*1854: Fashion plates, groups of ppl: File:Paris, New York &amp; Philadelphia fashions for spring and summer 1854, published &amp; sold by F. Mahan, no. 211 Chestnut Street, Philadelphia. LCCN2003655770.jpg
*c. 1855, French fashion plate, frock coats, waistcoats, etc.: File:Album z ilustracjami do czasopism z moda francuska, niemiecka i angielska. ca 1855 (9109644).jpg
* 1857, cream, printed flowers on check, watch pocket: File:Waistcoat MET 1978.88.8 F.jpg
*
* c. 1860, dark floral? fabric with machine embroidery?: File:Waistcoat MET 1983.479.13 F.jpg
*
* late 1870s–early 1880s, gold, woven: File:Waistcoat MET 1979.34.1 F.jpg

=== Frock coats ===

* Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg (1866)[[File:Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg|thumb|Romanian (?) Politician, c. 1866, ill-fitting but probably handmade suit|center]]
Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.tif (right)[[File:Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.jpg|thumb|1860, American, young men with frock coats, performers??|center]]

* George Augustus Henry Sala.jpg (Mathew Brady, 1855–1865)
* [[File:George Augustus Henry Sala.jpg|center|thumb|George Augustus Sala, 1855–1865, LoC]]

==== Dress and Conduct Manuals, Histories, Victorian Theory, Catalogs ====
# [Baker, William Henry.]  ''Clothes Dictionary''. Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf.
# [[c:File:Hint's_about_men's_dress,_right_principles_economically_applied_(IA_hintsaboutmensdr00barr).pdf|Hint's about men's dress, right principles economically applied (IA hintsaboutmensdr00barr).pdf]] (book, conduct manual, American, 1888)
#[https://upload.wikimedia.org/wikipedia/commons/6/69/Fashion-_the_power_that_influences_the_world_%28IA_fashionpowerthat00foxg%29.pdf?utm_source=commons.wikimedia.org&amp;utm_campaign=index&amp;utm_content=original Fashion- the power that influences the world (IA fashionpowerthat00foxg).pdf] (1871)  (American, incl court dress, Victorian fashion theory, livery, &quot;White, as it is the color of day, is expressive to us of the cheerfulness or gaiety which the return of day brings.&quot; [145], &quot;Gentlemen's Furnishing and Goods,&quot; &quot;Head and Foot Covering,&quot; Index)

# [https://upload.wikimedia.org/wikipedia/commons/a/ae/How_to_behave_and_how_to_amuse._A_handy_manual_of_etiquette_and_parlor_games_%28IA_howtobehavehowto00sand%29.pdf?utm_source=commons.wikimedia.org&amp;utm_campaign=index&amp;utm_content=original How to behave and how to amuse. A handy manual of etiquette and parlor games (IA howtobehavehowto00sand).pdf] (1895; American; Men's Dress, p. 23; besides conduct and dress, it has parlor games)
# [https://upload.wikimedia.org/wikipedia/commons/8/8e/The_etiquette_of_men%27s_dress_%28IA_etiquetteofmensd00cluc%29.pdf?utm_source=commons.wikimedia.org&amp;utm_campaign=index&amp;utm_content=original The etiquette of men's dress (IA etiquetteofmensd00cluc).pdf] (1888; American; some ads; anthology of individually written articles; lots of sports, outdoor events; search for &quot;frock&quot; — cutaway vs Prince Albert)
#[https://upload.wikimedia.org/wikipedia/commons/e/e9/S.W._Silver_%26_Co.%27s_handbook_for_transatlantic_travellers_.._%28IA_swsilvercoshandb00silv%29.pdf?utm_source=commons.wikimedia.org&amp;utm_campaign=index&amp;utm_content=original S.W. Silver &amp; Co.'s handbook for transatlantic travellers .. (IA swsilvercoshandb00silv).pdf] (1883; American; lots of ads from SW Silver, with prices; has blank pages where ppl can write &quot;Introductions&quot; and &quot;Engagements,&quot; but this is not a conduct book, it's travel guide / catalog to London)
#[https://upload.wikimedia.org/wikipedia/commons/f/f6/How_we_go_to_those_who_cannot_come_to_us.._%28IA_howwegotothosewh00roge%29.pdf?utm_source=commons.wikimedia.org&amp;utm_campaign=index&amp;utm_content=original How we go to those who cannot come to us.. (IA howwegotothosewh00roge).pdf] (Roger, Peet &amp; Co.; American; catalog, spring 1891; pp. 11, clerical frock; 7 &quot;Prince Alberts&quot; double-breasted frocks; p. 23 shirts as underwear; has a thing on fitting and mail order; bathing suits, collars and cuffs; tennis shoes)

== Victorian Men's Fashion (Intro to the Men's Fashion Section) ==
Museums and other historical sites have plenty of women’s clothes from the nineteenth century. There is great variety in both outer wear and underwear. “Men’s garments of the Victorian period have survived in far less quantity than women’s.” Buck p. 185 More waistcoats have survived than any other garment. The hardest garment to find are trousers. Very few survived the nineteenth century. Buck p. 185 By 1850 trousers dominated the fashion for men; the trouser straps under the shoe had disappeared; and a fly front had replaced the fall-front. Buck 185

It took most of the nineteenth century for men’s fashion to evolve into the coat, trousers and vest (waistcoat) that dominated the twentieth century and appears to be unchanging in the twenty-first century. “Men’s clothing, like women’s, was specific to each occasion and type of activity during the late nineteenth century.” Payne2, p. 518 For example, formal wear for social events was different from the casual, everyday wear for upper middle class and aristocratic men. Victorians also dressed in ensembles for hunting and riding (included knickerbockers–breeches that had a fuller cut). Casual and looser clothing was worn when participating in sports. Many different cuts of coats were available, like the Norfolk jacket, the morning coat, or the sack coat.

&quot;Tailored wool garments, originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.&quot;&lt;ref name=&quot;:25&quot; /&gt; (458) 

Waugh: &quot;When, at the beginning of the nineteenth century, the guiding principle of English style in tailoring became concentrated in cut and fit, the inadequacy of the general practice of cutting by patterns was realized and tailors began their search for a system whereby they could draft out garments which would fit well and yet retain their style and elegance. This search led to a revolution in the art of cutting, and was probably sparked off by the invention of the tape measure early in the nineteenth century.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1683 of 2116)

Waugh: &quot;By the end of the eighteenth, [sic] century English tailors became the leaders of men's fashions, because their long experience and appreciation of the subtleties of cloth had developed their skill and they gave style and elegance to the practical country coats and so made them acceptable for fashionable wear.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1472 of 2116)


Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today.

Ready-made clothing affected the social structure, the economy, and the gender differentiation.


David Kutcha traces the origin of the 3-piece men's suit to King Charles II of England, who may be said to be wearing a coat, pants and a waistcoat, but otherwise is not wearing a &quot;suit&quot; we — or the Victorians — would recognize as such: &lt;blockquote&gt;... since 1666, male gentility has been associated with modesty and plainness in dress. Eschewing fashion as an increasingly feminized realm Charles II's vest inaugurated a new and essentially modern era of masculine aesthetics, one that reversed a long-held association between elaborate display and high social status. Manly thrift now displayed elite status.&lt;ref&gt;{{Cite book|title=The Three-Piece Suit and Modern Masculinity, England 1550–1850|last=Kutcha|first=David|publisher=University of California Press|year=2002|location=Berkeley and Los Angeles}}&lt;/ref&gt; (2)&lt;/blockquote&gt;So it's really Cromwell who should get credit??

The French Revolution caused both men and women to change stylistic direction radically. For one generation, stylish women wore simple white muslin dresses without foundation garments underneath. The changes in men's fashion was more lasting: men turned away from the ornamentation and ruffles of the ''ancien régime'', choosing instead understated and subtle tailoring to indicate good taste and wealth.

Poole

&quot;comfort and convenience&quot; — trousers, sack coats, collars and cuffs without a shirt, wool rather than silks and satins; knee breeches constantly riding up

men as leaders of fashion, in spite of our sense that men didn't do fashion in the way that women did, were more serious than that

=== Dandy Stuff ===
Beau Brummell: &quot;Men began to aspire to fine cutting, tailoring, and perfect fit in their clothes, flawless grooming and manners in themselves. The Englishman George Bryan (Beau) Brummell deserves much credit for the ideal of meticulous masculine appearance.&quot;&lt;ref name=&quot;:25&quot;&gt;{{Cite book|title=The History of Costume: From Ancient Mesopotamia Through the Twentieth Century|last=Payne|first=Blanche|last2=Winakor|first2=Geitel|last3=Farrell-Beck|first3=Jane|publisher=Addison-Wesley Longman|year=1992|isbn=0-06-047141-7|edition=2nd|location=New York, New York}}&lt;/ref&gt; (458) &quot;From 1796 to 1816, the Beau set the pattern for cleanliness and liberal use of starch.&quot; (459)

&quot;That society is bad whose members, however tenacious they be of forms of etiquette and elaborate ceremonials, have one code of manners for those whom they deem their equals, and another for those whom they esteem to be of less importance to them by reason of age, pecuniary condition, or relative social influence.&quot;&lt;ref&gt;Sandison, G. H., comp. ''How to Behave and How to Amuse: A Handy Manual of Etiquette and Parlor Games. In Two Parts''. The Christian Herald, 1895.&lt;/ref&gt; (69)

=== Gender Ideologies ===
'''Victorian gender ideologies''' relegated men to the &quot;public sphere&quot; and women to the &quot;domestic sphere.&quot; (??) More than that “the cult of youth . . . the secularization of sport and the influence of modern warfare in generating men’s fashions, the tendency to prefer modern ideas of comfort and convenience, and the transformation of fashion knowledge and fashion urbanism from print to hyperreality”&lt;ref name=&quot;:26&quot;&gt;McNeil, Peter. &quot;Men's Fashion: 1800–2022.&quot; Chapter 22. ''The Routledge History of Fashion and Dress, 1800 to the Present''. Routledge, 2024. https://opus.lib.uts.edu.au/bitstream/10453/182707/2/Men%27s%20Fashion%20200822_24_12_20_09_29_44.pdf

DOI: 10.4324/9780429295607-27.&lt;/ref&gt; (McNeil 1 or 411) changed the perceptions of masculinity and femininity.

=== Class ===
The growing middle class involved among other things more and more jobs and careers for young men as clerks, office workers, they were junior, subordinate, and they were commuting on the railroads from suburbs.
Carolyn Kirby:&lt;blockquote&gt;In western Europe the fashion for plain dark suits coincided with the rise of the affluent middle-classes in a world where the pace of industrialisation and the globalisation of trade was accelerating as never before. The sharp, dark business suit became the last word in male power-dressing. And so it remains to this day.&lt;ref name=&quot;:27&quot;&gt;{{Cite web|url=https://historiamag.com/invent-masculine-fashion/|title=The invention of masculine fashion|last=Kirby|first=Carolyn|date=3 December 2025|website=Historia: Magazine of the Historical Writers' Association|access-date=25 August 2026}}&lt;/ref&gt;&lt;/blockquote&gt;

=== Bertie ===
Albert Edward, Prince of Wales was very concerned with fashion and authoritative about it, with a very specific eye to small details. He was increasingly influential in men's fashion from the last third of the 19th and into the 20th century, and many noticed what he did and copied him.&lt;blockquote&gt;When that great lover of pleasure, Edward VII, visited Marienbad incognito as the Duke of Lancaster, he was followed by tailors from Paris, Budapest, Vienna, and Berlin who photographed him and took notes about his clothes. Edward VII introduced many [423–424] novelties into men’s fashion. ... He is also credited with making fashionable the creased trouser in 1909 (his groom dried them with a board weight after heavy rain, resulting in the line), turned-up cuff trouser (after hitching his trouser bottoms at a dirty racing track) and, as his girth grew, undoing the bottom button of his waistcoat.&quot;&lt;ref name=&quot;:26&quot; /&gt; (424)&lt;/blockquote&gt;Cowles&lt;blockquote&gt;The Prince was not just ‘a swell’. In the jargon of the day he was ‘a heavy swell’, and apparently there was a world of difference between the two terms. A swell was a rich young aristocrat who lived in extreme comfort; but a heavy swell added showmanship to the comfort and lived in a stylish luxury that even the French were obliged to envy. And of course the heavy swell was the acme of sartorial elegance.

The Prince did not mind changing his dress half a dozen times a day. He loved clothes, and since whatever he chose to wear became the prevailing fashion overnight, he soon was regarded as an expert on the subject. His tailor-in-chief made a fortune. For many years he patronised a Mr. Poole. He discovered this gentleman by accident. He went to the theatre one night to see a well-known actor by the name of Fecher playing ‘Robert Macaire’. As an impecunious adventurer [129–130] Fechter was obliged to wear a coat that was torn and dirty, but Bertie’s expert eye noticed the elegant cut. At the end of the performance he asked Fechter for the name of his tailor, and Mr. Poole’s future was assured.

The Prince had so many clothes he could never travel with less than two valets; and two more valets were left at home cleaning, brushing and pressing his vast wardrobe. There were suits and coats for every variation of every climate the world over. There were over a hundred pieces of headgear; and since Bertie was an honorary admiral and an honorary general of most of the countries of Europe, there was an entire room devoted to uniforms, sashes, epaulettes, belts, buckles, swords, feathers and other regalia. ...

Men’s clothes became of such importance that new [130–131] shops sprang up like mushrooms in Savile Row, Clifford Street and Bond Street. Most of the Prince’s innovations were inspired by comfort and convenience. He altered the cut of the evening dress waistcoat, he shortened the tails on the tail coat, he left his frock coat open (due to an increasing girth), he introduced the black homburg, and he attended race meetings, not in the frock coat hitherto ''de regueur'' but in tweeds. He tried having his trousers creased down the sides rather than the front and back, in order to hide his bandy legs, but this idea did not catch on, and he soon discarded it himself. But the prince was not the only arbiter of men’s fashions. The band of  &quot;heavy swells&quot; who followed his lead gave him plenty of competition. Lord Raglan and Lord Petersham invented coats which are still named after them. Lord Dupplin the dinner jacket and Lord Cardigan the button-up sweater. But Lord Hardwicke made the most spectacular contribution. Men’s silk hats were made of beaver which was left in its original rough, shaggy state. Lord Hardwicke polished his hat until he could see his face in it, and consequently was known as &quot;Glossy Top&quot;. He is responsible for the top hat as we know it today.&lt;ref name=&quot;:28&quot;&gt;{{Cite book|title=Gay monarch, the life and pleasures of Edward VII|last=Cowles|first=Virginia|publisher=Harper|year=1956|location=New York, New York|archive-url=https://archive.org/details/gaymonarchlifepl0000cowl/}}&lt;/ref&gt; (129–131)&lt;/blockquote&gt;
=== Men’s Clothing 1830–1900 (Intro to the Decades) ===

* “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.”&lt;ref name=&quot;:25&quot; /&gt; (458) Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. (this is middle-class men for day wear?)
* The silhouette for men sometimes followed the same line as women’s clothing on the top half of their bodies. For example, as women’s sleeves grew into the huge leg-of-mutton fullness, the shoulders of men’s coats also had added fabric to make a pouf at the shoulder, thus emphasizing a broad shouldered look across the front. Men’s silhouettes changed subtly without the exaggerated fullness of women’s skirts or the raising and lowering of the neckline. Once men began to wear trousers, they did not go back to formal knee breeches (except in European courts or for court wear for Victoria).
* The line of men’s garments changed in smaller increments than women’s. Coats and trousers might fit tighter in some decades and looser in others. They were slim, then boxy, flared, then straight and evolved into the contemporary line we see on men today.
* Men's fashions in the 19th century have been less researched than women's fashions, in part because of fewer artifacts of what men wore were saved in museums for study.
* Photography, the first century that we have actual images (rather than artists' interpretations) of what people wore.

==== Court and Formal Dress ====

* Livery, working class
* Formal dress was separate attire, used for specified occasions, varies by decade

==== Shirts ====

* Shirts and collars separated: &quot;by 1827 detachable collars became available.&quot;&lt;ref name=&quot;:25&quot; /&gt; (477) Shirts were considered more like underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries.
* Neck treatments (and shirt cuffs)
* Shirts were treated as underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries.

==== Coats and Jackets ====
The 1901 ''Clothes Dictionary'' says coats were &quot;The principal garment of men's attire.&quot;&lt;ref name=&quot;:30&quot; /&gt;{{rp|10}}

Several styles of coats were available to fashion-conscious aristocratic and upper-middle-class men. Men’s suits in this period were tailor made, fitted to one specific man. The factories, however, knew that the vast majority of men were working class and began to manufacture and sell ready-made shirts, coats and trousers. More clothes were available for working men and they cost less. The cheaper work clothes enabled upper-working-class men to own more than one set of clothes and to have more formal clothes for church or other events. Ready-made shirts were also available in generic sizes, small, medium and large. Generic sizing speeded up the mass production process and, since little of the shirts could be seen, they did not need custom fitting.

Almost all of the coats Victorian men wore could be single or double breasted, with a couple of exceptions like morning coats and Norfolk jackets.

On more formal occasions upper-class men wore a '''cutaway morning coat''' with light trousers during the daytime, and a dark '''tail coat''' and trousers in the evening. (define these terms) 

Morning coats was a general term for the jacket part of &quot;full dress&quot; or formal daytime wear, distinguished from formal evening wear. Most morning coats were cutaways, but the shape or cut is not the distinguishing feature: the bottom was curved or tailed, so the coat was short in front, it did not have a waist seam, single-breasted. Worn over a waistcoat and shirt.

* According to Payne, &quot;During the early 1820s tailcoats were the most popular type of coat.&quot;&lt;ref name=&quot;:25&quot; /&gt;{{rp|474}}
* &quot;Tailcoats gave way to skirted frock coats by the 1840s.&quot;&lt;ref name=&quot;:25&quot; /&gt; (491) Tailcoats came down as far as the waist and then had tails (or &quot;claw hammer&quot; tails) attached at the waist in the back. Single-breasted, could be worn buttoned or unbuttoned in the front, over a waistcoat and shirt.
* Nowadays, the only tailcoats are tuxedos, a term not in use until the 1880s, and named for Tuxedo Park, a village in New York populated by the elite. Originally only the white coat, ''tuxedo'' gradually came to mean the entire suit.
* By around 1840 &quot;The body of the coat had now five seams and was made from six separate pieces. This method of cutting the body was also applied to all nineteenth-century coats which had fitted bodies and separate skirts or tails.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1487 of 2116)
* &quot;The basic principle of cutting a dress-coat has not changed since 1840, but throughout its long life it has undergone many slight variations of proportion-length of tails, shape and size of collar, revers, waist high or low, etc.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1487 of 2116)

==== Frock Coats ====
[[File:Frock coat MET CI38.23.39 F.jpeg|alt=Old-fashioned coat hanging on a mannequin|thumb|Frock coat, 1840, American]]

* Frock coats were introduced in the 1820s and worn throughout the nineteenth century with only minor changes to the line or fit. Sleeves were full at the top and waists were tight, creating an hourglass form. Initially frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The rate of fashion change notable in women's fashion was true for men's clothing as well. The frock coat persisted for a century, but it was in a state of constant (but subtle) evolution, with collars and lapels, coat length, the waist seam, the fullness of the coat itself as well as the sleeve, color, patterns like stripes or plaids.
* A Prince Albert is a double-breasted, very well-tailored, form-fitting frock coat.&lt;ref name=&quot;:30&quot;&gt;[Baker, William Henry.]  ''Clothes Dictionary''.  Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf.&lt;/ref&gt;{{rp|33}}
* Gentlemen, clergy, military
* Men around the world and especially in the British Empire wore frock coats: images and descriptions exist of men wearing frock coats in India&lt;ref&gt;Woodyatt, Nigel Gresley. ''Under Ten Viceroys; The Reminiscences of a Gurkha''. London: H. Jenkins Limited, 1922. Pp. 31, 107, 1774.&lt;/ref&gt;, Iran (Persia at that time)&lt;ref&gt;Sparroy, Wilfrid. ''Persian children of the royal family; the narrative of an English tutor at the court of H. I. H. Zillus-Sultán, G.C.S.I''. London and New York, J. Lane, 1902. Pp. 11, 29, 30.[https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf?utm_source=commons.wikimedia.org&amp;utm_campaign=index&amp;utm_content=original https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf].&lt;/ref&gt;, Japan&lt;ref&gt;Watson, W. Petrie. ''Japan, Aspects and Destinies''. New York, Dutton: 1904. P. 7, 19, 31.&lt;/ref&gt;, Egypt and Turkey&lt;ref&gt;Butler, Alfred Joshua. ''Court Life in Egypt''. London: Chapman and Hall, 1887. P. 7.&lt;/ref&gt;, ...
* Payne: &quot;In the 1720s, Englishmen adopted another coat [than the redingote]: a straighter, functional style that originated among the working classes. Made of plain woollen cloth in practical colors such as brown, it had no pleats, narrow sleeves, a small flat collar, and a minimum of decoration. This became known as the frock coat and was useful for riding, hunting, and overseeing one's lands.&quot; (402)

==== Trousers ====
In 1872, in analyzing men's clothing as evolutionary behavior — like coloration for mating among birds in the natural world — George Darwin (Charles's son) says, of trousers:&lt;blockquote&gt;... a form yields to a succeeding form, which is better adapted to the then surrounding conditions; thus, when it ceased to be requisite that men in active life should be ready to ride at any moment, and when riding had for some time ceased to be the ordinary method of travelling, knee breeches and boots yielded to trousers.&lt;ref name=&quot;:33&quot; /&gt; (125 of 565)&lt;/blockquote&gt;

* Working-class men had been wearing trousers (or &quot;trowsers&quot;) increasingly since the 16th century in Europe,('''citation''') but after the French revolution the middle and upper classes adopted them because &quot;the safest dress was that of the working man, ... who did not wear knee breeches, but rather trousers.&quot;&lt;ref name=&quot;:26&quot; /&gt; (416)
* Lewandowski: &quot;First worn by the early horse-riding steppe people, introduced to Western world by Italians in 16th century. They were accepted as informal dress in the Directoire (1790–1815 C.E.). They became a part of semi-formal dress around 1815 and became worn commonly after 1830.&quot;&lt;ref name=&quot;:31&quot;&gt;{{Cite book|title=The Complete Costume Dictionary|last=Lewandowski|first=Elizabeth J.|publisher=Scarecrow Press|year=2011}}&lt;/ref&gt; (29)
* Breeches: Men’s bifurcated garment that ended just below the knee
* Trousers (or long pants): men’s bifurcated garment that went down to the ankles or foot
* The most sweeping change in men’s fashion before 1820 was in the length of their pants. Trousers that went from waist to foot were so much more comfortable and popular than knee breeches that they replaced what had dominated men’s wear since the sixteenth century. And trousers have been worn for over two centuries now.
* They were originally tailored but began to be mass produced. Although the aristocracy were required to wear knee breeches at court, they wore trousers more than they wore breeches, and they could afford to have their suits and shirts hand sewn and individually. Working class men had to contend with small, medium, and large from mass production.
* This bifurcated garment was modified in specific places, like the length, the fullness over the hips or the width of the leg. In summer trousers were lighter in color — &quot;grayish blue, aqua and pearl gray.&quot; (Payne 2, 505) In winter trousers mixed darker colors like &quot;steel gray and brown, black and brown, and black and green.&quot; (Payne 505) More on summer colors, winter colors?
* Breeches, before 1840: &quot;The front of the breeches was cut with flaps, or 'falls', 5 inches to 8 inches wide.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1547 of 2116)
* &quot;From 1840 all trousers were cut with a centre front fly.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1562 of 2116)
* [Find quote about QV's men's court dress and the men being so cold]

==== Waistcoats ====
For most of the century, the waist-length sleeveless coat that men wore over their shirt and under their suit coat was called a waistcoat. By the 1880s, everyday waistcoats were called vests, and what was called a waistcoat was worn with formal, evening attire like tailcoats and tuxedos. Necklines rose and fell and collars and lapels came and went over the century, but by the end of the century, formal waistcoats had low necklines and lapels.&lt;ref name=&quot;:25&quot; /&gt; (528) Many more waistcoats exist as artifacts in collections than vests, perhaps because they were special and thus were saved.[[File:Waistcoat MET CI60.22.55 F.jpg|thumb|Men's waistcoat, 1850s]]

Over the course of the century the waistcoat evolved from ornate and decorative to something simpler, whose fabric matched the suit coat and trousers — the vest, in fact. Victorian men put their creativity, color and specialty fabrics into their waistcoats. At the beginning of the century men's waistcoats were made from specialty fabrics, like velvet or brocade or a patterned silk. Early in the century, stylish men wore two vivid waistcoats at the same time.&lt;ref name=&quot;:25&quot; /&gt; (474) An elaborate waistcoat might have a colorful, patterned base fabric as well as a floral ribbon embellishment (for example, see right). Long after coats and trousers were made from the same dark wool, waistcoats continued to be colorful, making them the focus of the ensemble.

By the end of the 19th century, the plain style came to dominate men’s fashion (and continues to dominate it to this day). When waistcoats became vests, they became part of the plain style

==== Overcoats ====
As their name suggests, overcoats or surcoats were coats worn for warmth and for protection from rain and snow on top of the other articles of clothing, including suit coats and frock coats. Overcoats became a significant fashion element, and many styles were seen throughout the century. [Top coat]

Victorians used the terms ''overcoat'' and ''great coat''., and sometimes perhaps ''surtout''. A great coat was a &quot;large overcoat, commonly made of wool.&quot;&lt;ref name=&quot;:31&quot; /&gt; (124) In his 1901 ''Clothes Dictionary'', Baker says a surtout is &quot;An overcoat cut in the style of a frock coat.&quot;&lt;ref name=&quot;:30&quot; /&gt; (39) Lewandowski says something more generic — &quot;term referring to any man's long cloak or coat.&quot;&lt;ref name=&quot;:31&quot; /&gt; (282)

Victorians also specified which of the many kinds of overcoat: redingote, Chesterfield coat, Inverness coat, macintosh (or mackintosh), paletot, Ulster coat, Norfolk jacket
[[File:Dandies of 1831.jpg|thumb|Redingote on figure in back, 1831]]

===== Redingote =====
A redingote was a type of a &quot;full-length overcoat, usually open at the front and belted, popular in varying designs through the eighteenth and nineteenth centuries.&quot;&lt;ref name=&quot;:32&quot;&gt;Mitchell, Rebecca, ed. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.&lt;/ref&gt; (527 or 565) (The figure in the back in the fashion plate, right, shows a redingote from 1831.) Because the redingote was popular for so long, it existed in various forms through the two centuries. Generally, it was single- or double-breasted, with a split skirt or box pleats in back (or both) and with a high standing collar, a rolled collar or a double collar. Derived &quot;from riding coat,&quot; (Payne 402) it was fitted at the waist and reached mid-calf, with sleeves that were full at the shoulders and upper arm and narrow from the elbow to the wrist, sporting a wide cuff. The length varied over time between knee- and ankle-length.

Also, women wore redingotes — in France (1865–1890) — that were shaped  to cover the bustle.&lt;ref name=&quot;:31&quot; /&gt; (247–248) Redingotes continued to develop, eventually evolving &quot;into an overcoat ..., but the overcoat lacked the pleats and large cuffs of the redingote,&quot;&lt;ref name=&quot;:25&quot; /&gt; (402) as well as, possibly, the waist seam.[[File:George Stanhope, 7th Earl of Chesterfield.jpg|thumb|George Stanhope, 7th Earl of Chesterfield, wearing a Chesterfield coat, 1860]]

===== Chesterfield Coat =====

* &quot;Chesterfield coats were single-breasted, with fly fronts [regular buttons and button holes, as opposed to fall front, in which a panel of cloth covers the button holes and the placket covers the closure completely] and velvet collars.&quot;&lt;ref name=&quot;:25&quot; /&gt;  Payne2 p. 519
* They were one kind of overcoat, named for George Stanhope, 6th Earl of Chesterfield. 1840s.
* Signature velvet collar.
* &quot;Man's fitted overcoat with hidden buttons and velvet collar&quot;&lt;ref name=&quot;:31&quot; /&gt; (62)
* &quot;fitted men’s knee-length overcoat, narrower than the frock coat and without a waist seam, popular from the end of the nineteenth century.&quot;&lt;ref name=&quot;:32&quot; /&gt; (519 of 565)

===== Ulster Coat =====
In 1872, in analyzing men's clothing as evolutionary behavior — like coloration for mating among birds in the natural world — George Darwin (Charles's son) says, of Ulster coats:&lt;blockquote&gt;... a form yields to a succeeding form, which is better adapted to the then surrounding conditions .... The &quot;Ulster Coat&quot;, now so much in vogue, is evidently largely fostered by railway travelling, and could hardly have flourished in the last century, when men either rode or travelled in coaches, where there was no spare room for any very bulky garment.&lt;ref name=&quot;:33&quot;&gt;Mitchell, Rebecca, ed. &quot;3. 'George H. Darwin, ‘Development in Dress', ''Macmillan’s Magazine'' 26 (September 1872), pp. 410–16.&quot; Section 1: Fashion Theory in the Nineteenth Century. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.&lt;/ref&gt; (125 of 565)&lt;/blockquote&gt;

* &quot;men’s overcoat with attached waist-length cape, popular in the nineteenth century.&quot;&lt;ref name=&quot;:32&quot; /&gt; (530 of 565)
* ulster overcoat for travel.
* &quot;Man's single- or double-breasted overcoat with waist belt. Initially had hood, but by 1870s was detachable. In 1875, the ticket pocket was added on the left sleeve above the cuff.&quot;&lt;ref name=&quot;:31&quot; /&gt; (304)
* &quot;Travel called for an ulster — a long, loose, heavy overcoat — or for an Inverness coat with attached cape.&quot;&lt;ref name=&quot;:25&quot; /&gt; (527)
* &quot;Heavy tweed coats for travel such as the ulster had half-belts and box pleating in back.&quot;&lt;ref name=&quot;:25&quot; /&gt; (551)
* Ulster may have had a shorter cape than the Inverness
* Can we find a source linking John McGee (of McGee &amp; Co.) with the invention of the Ulster coat in 1866? (per Wikipedia)?
[[File:The man wot drove the sovereign to the last stage trying it on the next. (BM 1868,0808.9200).jpg|thumb|Duke of Wellington, 1830, wearing a coachman's coat, or Inverness coat]]

===== Inverness Cape =====

* &quot;men’s overcoat, with long, full, open sleeves, giving a cape-like appearance from the front; worn from the mid-nineteenth century to the early twentieth century as formal wear.&quot;&lt;ref name=&quot;:32&quot; /&gt; (523 of 565)
* an Inverness coat with up to five cape collars and a high curved collar.
* Lewandowski: &quot;(1840–1865 C.E.) to 20th century. Worn after 1858, man's large, loose, knee-length overcoat with fitted collar and long cape.&quot;&lt;ref name=&quot;:31&quot; /&gt; (143)
* An Inverness coat had a high standing collar as well as up to 5 capes over a knee-length, loose overcoat.&lt;ref name=&quot;:31&quot; /&gt; (143)
* Caricature of the Duke of Wellington in a &quot;coachman's caped great-coat,&quot;&lt;ref&gt;{{Cite web|url=https://www.britishmuseum.org/collection/object/P_1868-0808-9200|title=Duke of Wellington in Coachman's coat. print; satirical print {{!}} British Museum|website=www.britishmuseum.org|language=en|access-date=2026-11-04}}&lt;/ref&gt; or Inverness, 1830
* In the 1880s, the Inverness was sometimes sleeveless, so the capes were not merely decorative.
* both formal and informal styles exist; also called a coachman's cape.

[[File:Carson, Pirie, Scott &amp; Co. Macintosh, 1893.jpg|thumb|360x360px|Gentleman's Macintosh, 1893]]

===== Mackintosh =====

* &quot;Introduced in 1836, short loose overcoat made of Mackintosh's [sic] patent India-rubber cloth. Came in drab or dark green.&quot;&lt;ref name=&quot;:31&quot; /&gt; (181)
* &quot;waterproofed overcoat, named after its inventor [Charles Macintosh]; now used generically to refer to raincoat.&quot;&lt;ref name=&quot;:32&quot; /&gt; (524 of 565)
* [according to Wikipedia, the spelling ''Mackintosh'' is now standard.]

===== Paletot =====

* &quot;short, loose overcoat for men or women.&quot;&lt;ref name=&quot;:32&quot; /&gt; (526 of 565)
* &quot;The pilot coat, or paletot, was the original of all the loosely cut garments which appeared from the 1830's onwards and which inspired the more casual coats and sportswear of the end of the century.&lt;ref name=&quot;:34&quot;&gt;{{Cite book|title=The Cut of Men’s Clothes: 1600-1900|last=Waugh|first=Norah|publisher=Theatre Arts Books|year=1987|isbn=978-0878300259}}&lt;/ref&gt; (1533 of 2116)

===== Norfolk Jacket =====

* &quot;A single breasted sack coat having two or three box plaits [pleats] in back, two in front and a belt of material. Most used for hunting and outdoor sports.&quot;&lt;ref name=&quot;:30&quot; /&gt; (28)
* &quot;A variation 'suitable for any kind of outdoor exercise', was the 'Norfolk jacket' which appeared in the 1870's. It also was a two-seamed jacket but was cut full and the fullness arranged in box pleats, two in the front and two in the back. It had a belt and was worn with 'knickerbockers'-knee breeches with legs cut loose and gathered into a knee-band.&quot;&lt;ref name=&quot;:34&quot; /&gt; (1547 of 2116)


&lt;ref name=&quot;:24&quot; /&gt;Kay, Fiona and Neil R. Storey, ''Victorian Fashions for Women'', Pen &amp; Sword History, Yorkshire - Philadelphia, 2022

[[File:Homosexualitywilde.jpg|thumb|Oscar Wilde and Bosie Douglas, straw boater and derby, 1894]]

==== Hats ====
The upper-class continued to wear top hats, and [[bowler hats]] (known in America as a derby) were worn by the working class. 1830s This class of men also wore round cloth caps, often seen on male children.

The straw boater was the most popular hat by the end of the century and was worn by all of the classes.

Men wore [[top hats]], with wide brims in sunny weather. 1830s - 1900

Top hats briefly became the very tall &quot;stovepipe&quot; shape, but a variety of other hat shapes were popular.

Homburgs, deerstalkers and “Trilbies” were also popular. Buck, p. 198-199 (George de Maurier novel ''Trilby'') Hats were ''de rigueur'' for all Victorians, all classes, all ages, all genders.

==== Shoes ====
[[File:Men's fashion silhouette of 1837.jpg|thumb|Men's coats, 1837]]
[[File:Lev Wittgenstein by F. Kruger.JPG|thumb|Lev Wittgenstein by F. Kruger, 1836, wearing a surtout]]

=== 1830s and 1840s ===
For men's fashion, the 1830s and 1840s were dominated by a defined waist, lean limbs and tall hats.

Working-class men had been wearing trousers for some time, but by the 1830s and '40s, middle- and upper-class men generally wore frock coats and trousers for casual or less formal attire in the city. In the country, what they wore depended on what they were doing — hunting, riding, shooting and so onen[[File:Man's Frock Coat LACMA M.2007.211.61 (1 of 4).jpg|alt=Male mannequin wearing a light-colored thigh-length coat similar to a suit jacket and a top hat|thumb|Men's frock coat, c. 1845]]
[[File:Stockholms mode-journal- Tidskrift för den eleganta werlden 1847, illustration nr 2.jpg|thumb|Stockholm's fashion magazine, 1847]]Some men wore corsets during this time, so the waist was nipped in and the skirt of the frock coat flared out. The sleeves of their coats and jackets were slightly puffed at the shoulders. The trouser legs were straight with a strap under the instep. The figure on the left in the fashion plate from ''Modes de Paris Journal des Tailleurs'' (right) illustrates this silhouette very clearly as well as the frock coat, the strap under the instep, and the puffed shoulders.

==== Coats and Jackets ====
Frock coats had become fashionable for day wear in the 1820s&lt;ref name=&quot;:25&quot; /&gt;{{rp|474}} and were worn throughout the century. In the 1830s frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. 

The main changes in frock coats in these two decades were in the colors and the neck treatments (collar and lapels). Initially frock coats were not necessarily black. They could be colorful — like &quot;plum, mulberry, brown, bright blue, and bottle green&quot; — with &quot;contrasting velvet collars.&quot;&lt;ref name=&quot;:25&quot; /&gt;{{rp|474}} The truly bright, saturated colors made possible by synthetic dyes, however, were a couple of decades away still and were never used for frock coats. 

Middle- and upper-class men limited their use of tailcoats to evening wear and more formal occasions. (In the 1847 Stockholm fashion magazine, below right, the figure in the center is wearing a tailcoat.) A cutaway morning coat was worn with light trousers for formal events during the daytime, and a dark tail coat and trousers were worn in the evening.

==== Shirts ====
Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[cravats]] or neck ties. Before they were mass produced, shirts made of finer linen or cotton were tailored for upper and wealthy middle-class men. Depending on occupation, working-class men's shirts were made from a rougher, home-woven linen or cotton, often shaped loosely like a smock and sometimes without a front closure. Shirts were sometimes as long as knee-length. 

==== Trousers ====
Men had adopted trousers over knee breeches for semi-formal dress &quot;around 1815&quot;&lt;ref name=&quot;:31&quot; /&gt; (298) beginning after the French Revolution.&lt;ref name=&quot;:26&quot; /&gt; (416) They were wearing trousers both for formal and semi-formal dress by the 1830s. Initially trousers &quot;fit quite snugly ..., but in the 1840s they became looser.&quot;&lt;ref name=&quot;:25&quot; /&gt; (477)

Upper-class and aristocratic men still wore knee breeches at court and loose-fitting breeches (in the form of knickerbockers) for country wear and some sports, like riding and golf. Generally, however, they wore trousers more than they wore breeches.

Men flocked to them once trousers became a stylish option because the looser fit was warmer in cold weather and more comfortable than the tight-fitting knee breeches from earlier in the century.

Ready-made men's clothing was available in the US early, but by the 1840s ready-made suits were making an impact in Europe and the UK,&lt;ref name=&quot;:26&quot; /&gt; (419–420) especially outside the upper classes. Aristocrats and upper middle class could afford the well-tailored cut and fit of individually hand-sewn trousers and coats.

==== Waistcoats ====
Waistcoats were both single- or double-breasted, with shawl or notched collars, and might be finished in double points at the lowered waist. Collars gradually lowered over the course of the two decades. Often the colorful vests were decorated with fur or velvet collars.

“Waistcoats remained colorful, decorative items in men’s wardrobes after the rest of their costumes had become plain.” (Payne 2 460-462)  

&lt;nowiki&gt;:&lt;/nowiki&gt; &quot;During the 1820s and 1830s, men sometimes wore two waistcoats, in combinations like white velvet over rose-and-gold brocade.&quot;&lt;ref name=&quot;:25&quot; /&gt; (474)  

==== &quot;Overcoats ====
Men wore a variety of topcoats or overcoats. Surtouts were overcoats built like long frock coats, like the one worn by Lev Wittgenstein in the portrait (above right). The redingote was most popular in the 1830s and 1840s. 

==== Original Text ====
During the [[1840s in fashion|1840s]], men wore tight-fitting, calf length [[frock coat]]s and a [[waistcoat]] or vest. Sleeves were full at the top and waists were tight, creating an hourglass form. For more formal occasions, a cutaway morning coat was worn with light trousers during the daytime, and a dark tail coat and trousers was worn in the evening. Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[Cravat (early)|cravat]]s or neck ties. Men wore [[top hat]]s, with wide brims in sunny weather.
=== 1850s ===
The 1850s revealed several industrial inventions that significantly impacted clothing. First, the Industrial Revolution began (and still does today) with machines to weave fabric and machines to sew fabrics. Manufacturers quickly learned that the majority of the population was working class. Theirs were the simplest, least trimmed garments made in the nineteenth century. Second, mass production of shirts, trousers and coats changed how most people shopped for clothes. The aristocracy and upper middle class folks could still have their clothes made to specifically fit their bodies. Manufacturers knew that they could not mass produce clothing for each individual’s body. They invented generic sizing in small, medium, and large. They could, therefore, produce many more items of clothing with generic sizing. Third, Isaac Singer patented and began to manufacture home or domestic sewing machines (treadle machines) enabling more form-fitted '''dresses''' to be constructed for an individual’s specific shape and size. The fourth item to impact fashion and clothing was the invention of the paper pattern. These were introduced in fashion magazines targeting women. Almost all women were taught to sew by hand, whether they were aristocrats or working class. Sewing machines enormously speeded up the '''process of constructing from hours to minutes'''. Finally, aniline dyes made colors brighter and longer lasting. Prints and stripes, checks and plaids, fabrics had much more vibrant color with aniline dyes. The Industrial Revolution impacted the working class and changed their lives the most.

children's clothes? Women made men's clothing too
According to Judith Flanders, working-class men's clothes could be colorful. For example drivers of hansom cabs () dressed more up-scale, and drivers of hackneys () did not.&lt;blockquote&gt;While hackney drivers were also considered to be stereotypically shabby, hansom-cab drivers were generally represented as smartly dressed. A print in 1850 showed a driver in a snappy brown coat instead of the coachman’s heavy multiple-caped outfit, pale green striped trousers, short boots and top hat, the [167–168] reins held daintily in his gloved hands. Both cab and coach drivers wore top hats, but cabbies of a sporting bent later switched to bowlers, and in summer donned bright checked outfits.&lt;ref name=&quot;:23&quot;&gt;{{Cite book|title=The Victorian City: Everyday Life in Dickens' London|last=Flanders|first=Judith|publisher=Thomas Dunne Books|year=2012|location=New York, New York}}&lt;/ref&gt; (167–168 [of 972])&lt;/blockquote&gt;

A variety of silhouettes was seen in the 1850s. Trousers were long and slim with a strap under the instep of the shoe. 

==== Coats and Jackets ====
&quot;Although the frock coat was worn throughout the century, it lost its dominant position in the 1850s, when the morning coat began to replace it.&quot;&lt;ref name=&quot;:29&quot; /&gt;

A jacket shorter than the frock coat or morning coat that had visible pockets began to be worn in about 1850. Cutaway tail coats were still reserved for formal occasions. All of these coats could be single or double breasted. They were called “ditto suits” by those who wore them because the coat, trousers, and waistcoat were all made from the same fabric. There were morning coats and cutaway morning coats; there were Norfolk jackets and ditto suits. The most formal were the cutaway tail coats for evening wear and balls; and the least formal were the boxy shaped sack coats. By the 1860s, it had become the most popular casual coat.

==== Shirts ====
During the [[1850s in fashion|1850s]], men started wearing shirts with high upstanding or turnover [[collar (clothing)|collars]] and [[necktie#Four-in-hand|four-in-hand necktie]]s tied in a bow, or tied in a knot with the pointed ends sticking out like &quot;wings.&quot;

==== Trousers ====
Trousers had fly fronts and breeches were used for formal functions and knickers (short pants fuller cut than breeches) for horseback riding. Men's choice to wear trousers is an instance of the importance of comfort and convenience outweighing style and presentation, a value that increased in importance through the 20th century.

==== Waistcoats ====
Waistcoats remained the one garment that could be brightly colored and copiously trimmed on exotic fabrics. Waistcoats show that men were concerned with decorative elements in their ensemble.

==== Notes &amp; Quotes ====

* transition from frock coats to ditto suits, 1850s (Payne, 463)
==== Hats ====
The upper-class continued to wear top hats, and [[bowler hat]]s were worn by the working class. Caps

==== Overcoats ====
Overcoats were cut fuller with wider sleeves and no extra fullness at the shoulder. Unlike the redingote, the overcoats had straight seams to the knee length. Many had wide lapels that reached below the waist. Working class men and their families benefitted more from the industrial revolution in regards to clothing.  Manufacturers marketed to them because because they were the largest class.
[[File:Gazette of Fashion, February 1865, RP-P-2009-2675.jpg|thumb|1865, Frock coats, etc. (Paris)]]
[[File:Gazette of Fashion, October 1864, RP-P-2009-2673.jpg|thumb|1864, Frock coats, etc., Paris]]
[[File:Broadway fashions, New York LCCN2003677917.jpg|thumb|Butterick, Spring-Summer 1866, Broadway Fashions]]
[[File:Four young men wearing frock coats - DPLA - 4706a21c9267c45034d51c8fc5309251 (page 1).jpg|thumb|1860s, working class?, ill fitting, frock coats, trousers, shirts, vest, derby hats]]

=== 1860s ===
[[File:Corporal frock coat 1862 med.jpg|thumb|Corporal frock coat 1862, 9-botton military]]The 1860s saw fewer inventions that influenced or changed the way people were acquiring clothes. But the changes begun by mass production continued with the working class. Little changed for the higher classes, except maybe paper patterns and sewing machines.

The different shape and size in clothing continued to provide slightly different silhouettes for men in the upper and middle classes. As women's clothes grew more complex in cut and style, men's clothes became simpler in cut and style.

==== Coats and Jackets ====
Individualized fittings continued for upper class and wealthy middle class clients. (Working class as well as middle and upper class.) Working class was well over 60% of the population. For the largest class, ready-made garments were ideal for men and women who were working class. 

For the two upper classes morning coats still led the way but frock coats were still in demand and worn in business and more casual situations. Leisure wear like knickers and ditto suits became more popular in the 1860s. Haute couture wanted every occasion or event to have special clothing. In some ways men dressed more alike (coat, trousers, waistcoat) and in some ways there was more variety in the cut of their coats and in the manufactured clothing they could buy. Many upper class men still had their shirts custom made.

Frock coats were shortened to knee-length and were worn for business, while the mid-thigh length [[sack coat]] slowly displaced the frock coat for less-formal occasions, with the overall effect of a looser silhouette.

The Prince of Wales: &quot;For the countryside such as at Sandringham, he permitted an informal dress code. The Henry Poole ledger marked as 'HRH 1865' is for an evening coat without tails, the first 'dinner jacket.'&quot;&lt;ref name=&quot;:26&quot; /&gt; (424)

==== Shirts ====
Ready-made shirts rarely were seen since shirts were more underwear than outerwear. Factories began to make and sell detachable collars and cuffs. It was more convenient to replace collars and cuffs each day than the time it took a servant (working class) to wash and starch enough to look like the celluloid many of those collars were made of. 

==== Trousers ====
Trousers were cut full and boxy during most of the 1850s and 1860s. The strap under the shoe disappeared and trouser legs reached the back of the heel in length. Knickers were still worn for hunting or riding and breeches were still ''de rigueur'' at court.

==== Waistcoats ====
Waistcoats remained decorative and worn with most of the coats that were popular. The young “dandies” wore the most colorful and decorated waistcoats. These “dandies” gave their highest priority to their appearance. They were the avant garde of haute couture for men.

==== Necks Treatments ====
In the [[1860s in fashion|1860s]], men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin.

==== Hats ====
Top hats briefly became the very tall &quot;stovepipe&quot; shape, but a variety of other hat shapes were popular.

==== Overcoats ====
The single-breasted redingote continued in popularity among the aristocracy and upper middle-class.  These coats were often “in colors such as plum, mulberry, brown, bright blue, and bottle green . . . [with] contrasting velvet collars.” Payne2 p. 474 [[File:Mens Coats 1872 Fashion Plate.jpg|thumb|upright|Drawing of Victorian men 1870s]]
[[File:Ashurst frock coat.jpg|thumb|Ashurst Frock coat]]
[[File:Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat; Standing (fbc266c2feca43f6ad5b44e031635a8d).tif|alt=Old photograph of a man wearing a dark thigh-length coat similar to a suit coat with light-colored trousers|thumb|Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat, c. 1872]]
[[File:Caillebotte Hugot.jpg|thumb|Portrait of Paul Hugot, 1878]]
[[File:Oscar Wilde (1854-1900), by Hills &amp; Saunders, Rugby &amp; Oxford 3 april 1876.jpg|thumb|Oscar Wilde, 1876, ditto suit]]
[[File:FrockCoat.jpg|thumb|English frock coat, 1870s]]

=== 1870s ===
To correct and prevent errors being made in court dress, in 1875 the Lord Chamberlain published ''Dress Worn by Gentleman at Her Majesty's Court'', &quot;a summary of regulations for court uniform and dress.&quot;&lt;ref&gt;{{Cite book|url=https://www.google.com/books/edition/Dress_worn_by_Gentlemen_at_Her_Majesty_s/pvrbQCXq0MEC?hl=en|title=Dress worn by Gentlemen at Her Majesty's Court|last=Britain)|first=Victoria (Queen of Great|date=1875|language=en}}&lt;/ref&gt; This is the kind of thing Bertie really cared about. Define court dress.

A portly, boxy silhouette was sought for most of the 1870s. Older men whose stomachs had spread outward and thus looked prosperous to the Victorians were the ideal for men’s high fashion. Sack coats and trousers were cut a little wider in the legs.

Corduroy for trousers and muslin for shirts was often used in working class clothing.

==== Coats and Jackets ====
There were minor changes to some of the coats Victorian men wore. Frock coats were shortened to the knee. Sack coats (or ditto coats — the forerunner of the suits worn by men in the western cultures of the twentieth and twenty-first centuries) were found to be so comfortable that most men of the upper and middle classes had a ditto suit. Men still wore morning coats and the aristocracy still wore cutaway coats with trousers to parties and balls but cutaway coats with knee breeches at court.

==== Shirts ====
Although ready-made shirts were available, the upper and middle classes still wore bespoke clothing hand-made, with better fabric and an excellent fit but stitched on machines. The suits and shirts made for the working class were of cheaper fabrics.

==== Trousers ====
Most men were wearing trousers  all of the time (except at court) and have worn them ever since. Trousers had button flies by the 1870s which most men found to be much more convenient than laces. Trousers did not “ride up” the leg in the way that knee breeches did when dancing or sitting. If a man did not have fashionable legs, they could be obscured behind trousers.

==== Waistcoats ====
White waistcoats began to be worn by men. Black were also worn when mourning. Changes in the waistcoat could be seen but were very subtle. For example, collars were lowered as the neckline grew deeper and deeper in front, revealing more of the fancy vest.

==== Overcoats ====
A longer version of the standard overcoat, “sometimes with fur collar, cuffs, and lining” was often worn for warmth. Payne2 p. 519

==== Original Text ====
During the [[1870s in fashion|1870s]], three-piece suits grew in popularity along with patterned fabrics for shirts. Neckties were the four-in-hand and, later, the [[Ascot tie]]s. A narrow ribbon tie was an alternative for tropical climates, especially in the Americas. Both frock coats and sack coats became shorter and more form fitting. Flat straw boaters were worn when boating.
[[File:Elliott &amp; Fry10a.jpg|thumb|William Gladstone, 1887, morning coat]]
[[File:British politicians gambling at Ascot Wellcome V0050378.jpg|thumb|British Politicians Gambling at Ascot, 1884]]
[[File:Scribner's magazine (1887) (14595087238).jpg|thumb|1887, types of American servants]]

=== 1880s ===
The silhouette narrowed and straightened. That is, the line went straight down without the flare of skirts on the coats. This smooth line had few decorations. Men wore hardly any jewelry. They might have a boutonniere or a ring.  Suits were not  decorated. All coats were straighter and shorter.

Depending on the time of day, the Prince of Wales might change his clothes (thus, his silhouette) up to six times per day,&lt;ref name=&quot;:28&quot; /&gt; (129) and some aristocrats and upper-class men would follow his lead.

==== Coats and Jackets ====
Although the frock coat was worn throughout the nineteenth century, the single fabric sack suit was worn for casual wear and was beginning to be seen as a businessman’s suit.  Morning coats were still in use for some events as well as casual occasions. The modern tuxedo or Cowes jacket appeared and was worn for formal occasions. The tuxedo is still worn as formal dress today.

==== Shirts ====
Shirts were still custom made for the rich and the working class bought ready-made shirts. The shirt remained basically the same but it could handle the different neck treatments.  Even mass produced shirts could have buttons or button holes for detachable collars and cuffs.

==== Trousers ====
Trouser legs began to decrease in width as the coat became narrower and emphasized strong vertical lines.

==== Waistcoats ====
During the 1880s waistcoats were called vests [is it regional?] and much of their bold trim and colorful fabric disappeared and vests were constructed from the same fabric as the coat and trousers. 

The business suit for men developed in the 1880s and was firmly established in the 1890s.

==== Overcoats ====
Versions of topcoats increased, adding an ulster overcoat for travel and an Inverness coat with up to five cape collars and a high curved collar. Double breasted topcoats were popular and some were knee-length and some calf-length.

==== Original Text ====
During the [[1880s in fashion|1880s]], formal evening dress remained a dark tail coat and trousers with a dark waistcoat, a white bow tie, and a shirt with a winged collar. In mid-decade, the dinner jacket or [[tuxedo]], was used in more relaxed formal occasions. The [[Norfolk jacket]] and tweed or woolen breeches were used for rugged outdoor pursuits such as shooting. Knee-length topcoats, often with contrasting velvet or fur collars, and calf-length overcoats were worn in winter. Men's shoes had higher heels and a narrow toe.
[[File:American fashions - The Jno. J. Mitchell Co. Litho., New York. LCCN2006676665.jpg|thumb|Fashion plate from the Jno. J. Mitchell Co., New York, 1899]]
[[File:Alfred Milner Vanity Fair 1897-04-15.jpg|thumb|Alfred Milner, 1897, Leslie Ward (&quot;Spy&quot;), for Vanity Fair]]
[[File:Nobl-08.jpg|thumb|Sherlock Holmes, 1892, Sidney Paget]]
[[File:PUK1894page380rector poor.jpg|thumb|Rector and Drinker, 1894]]

=== 1890s ===
The dominant silhouette for men became the sack suit in the 1890s and while changes in the sack have been subtle, it remains the dominant silhouette today.

==== Coats and Jackets ====
“Sacks became ''the'' favored style of suit coat,”(Payne2, p. 532) especially among the younger men. Frock coats were associated with older men along with the cutaway coat. There were many more choices for sports wear and leisure wear than ever before. Norfolk jackets were often worn while hunting. Most informal styles could be purchased ready made. The rich still had their clothing custom made by tailors but even working class men could purchase most ready-made versions of the coats, trousers, and vests.

==== Shirts ====
Shirts could be made of patterned fabric as well as solid colors and white.  Shirts were beginning to be acceptable as outer wear especially when worn as sporting clothing.

==== Trousers ====
Early in the 1890s, “slim, straight trousers were news.” (Payne 2, p. 532) Trousers could be plaid or checked as well as the same fabric as the coat and vest.

==== Waistcoats ====
Fancy fabrics and bold trim were not seen much as the more subdued vest dominated the fashion scene.

==== Neck Treatments ====
In the 1860s, men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin.

==== Overcoats ====
“Topcoats included fingertip-length styles made of twill covert cloth. Single-breasted, fly-front overcoats were fashionable too. Roomy overcoats with fur linings and collars warmed the wealthy. Inverness overcoats remained in vogue.” Payne2, p.532

==== Original Text ====
Starting from the [[1890s in fashion|1890s]], the [[blazer]] was introduced, and was worn for sports, sailing, and other casual activities.&lt;ref&gt;{{cite web|last=Landow|first=George|url=http://www.victorianweb.org/art/costume/90s/2.html|title=Men's informal sporting dress, late 1880s and '90s}}&lt;/ref&gt;

Throughout much of the Victorian era most men wore fairly short hair. This was often accompanied by various forms of facial hair including moustaches, side-burns, and full beards.  A clean-shaven face did not come back into fashion until the end of the 1880s and early 1890s.&lt;ref&gt;{{cite web|url=http://www.victorianweb.org/art/costume/nunn21.html|title=Victorian Men's Fashions, 1850–1900: Hair}}&lt;/ref&gt;

Distinguishing what men really wore from what was marketed to them in periodicals and advertisements is difficult, as reliable records do not exist.&lt;ref name=&quot;shannon597&quot;&gt;{{cite journal|last=Shannon|first=Brent|title=Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914|journal=Victorian Studies|year=2004|volume=46|issue=4|pages=597–630|doi=10.1353/vic.2005.0022}}&lt;/ref&gt;

=== Notes ===

*Men's suits buttoned higher up than today (Payne, 467)
* Norfolk jackets and sack suits (Payne, 471)
* formal attire, tuxedos with tails, cutaways (Payne, 469)
* Keith Middlemas (https://archive.org/details/storyoffiesta00huxf/page/200/mode/2up?q=fashion)
*Brent Shannon. &quot;Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914.&quot; Victorian Studies 46, no. 4 (Summer 2004): 597–630.

== Hats and headwear ==
[[File:Ford.madox.brown.last.emma.study.jpg|thumb|''Emma Hill'' by [[Ford Madox Brown]] (1853), a woman wearing a later version of the [[poke bonnet]]]]
[[File:Hoed,_objectnr_KA_1237.tif|left|thumb|Perched bonnet style of the early 1870s.]]
Hats were crucial to a respectable appearance for both men and women. 

=== Men's Hats ===
The top hat, for example, was standard formal wear for upper- and middle-class men.[Payne] According to Blanche Payne, &quot;The high top hat, usually black or dark gray, had reached its characteristic shape by 1798 and dominated the entire nineteenth century.&quot; (457–58) 

Although top hats were the dominant hat in the 19th century, other hats became popular for working classes and lower income middle class. “The style of an individual’s hat varied, depending on fashion and their social position, as well as their profession or chosen activity..” ''Goodman 53 of 460'' Other hats that became necessary and popular include the Derby, the straw Boater, and a flat cap with a short brim. In many cases the class, work activity and income could be determined by what kind of hat was on the head of the wearer. For some men, a hat supporting a particular sport or team was important.

The Derby or Bowler hat was designed by William and Thomas Bowler, brother shopkeepers in 1849. ''(Goodman 55 of 460)'' It cost less than a top hat but  lasted longer and was soon worn by middle class bankers and clerks. Straw boaters were worn by the aristocracy for casual events and working class factory workers and agricultural laborers. By 1901 working class men had changed their preference to the flat caps which became the most popular hat for the workers. 

Headdress for men was an essential part of dress for the entire period of the Victorian age, from the 1830s through the end of the century. Judith Flanders describes the hats worn by men in London, &lt;blockquote&gt;It is difficult to bear in mind the importance of hats as not only markers of class and income, but also as indicators of respectability. [509–519] [George Augustus] Sala commented that &quot;every&quot; man throughout the history of the world &quot;must, necessarily and habitually, wear some kind of covering to his head&quot;. Postmen wore hats, small children wore hats, field labourers and market gardeners wore hats, cricketers, skaters — all sportsmen — wore hats. It was, self-evidently, impossible to go outdoors without one. ... Those in professional occupations wore pot hats, as did clerks and all those with pretensions to middle-class status. Even doctors' delivery boys wore battered hand-me-down pot hats: &quot;the nap rusty, the band a mournful strip of tarnished lace; but still a Hat&quot;, which &quot;stamps him as being associated, in however slender a manner, with a learned profession&quot;. Cloth caps were for labourers, for costers and for boys. ... Artisans wore caps made out of paper, which they folded [510–511] themselves and so could easily replace as they became dirty.&lt;ref name=&quot;:23&quot; /&gt; (509–511 [of 972]) &lt;/blockquote&gt;A pot-hat is a kind of derby or bowler in men's hats. (Lewandowski, 237)

==== Original Text ====
Hats were crucial to a respectable appearance for both men and women. The top hat, for example, was standard formal wear for upper- and middle-class men.&lt;ref name=&quot;:4&quot;&gt;{{Cite book |last=Steele |first=Valerie |url=https://archive.org/details/fashioneroticism0000stee |title=Victorian Fashion. Fashion and Eroticism: Ideals of Feminine Beauty from the Victorian Era to the Jazz Age |publisher=Oxford University Press |year=1985 |isbn=978-0-19-503530-8 |pages=[https://archive.org/details/fashioneroticism0000stee/page/51 51]–84 |url-access=registration}}&lt;/ref&gt; For women, the styles of hats changed over time and were designed to match their outfits.

=== Women's Hats ===
For a discussion of the history of plumes and feathers, see [[Social Victorians/Victorian Things#Ostrich Plumes and Prince of Wales's Feathers|Ostrich Plumes and Prince of Wales's Feathers in ''Victorian Things'']].

==== Original Wikipedia Text ====
During the early Victorian decades, hats were modest in size and design, straw and fabric bonnets being the popular choice. [[Poke bonnet]]s, which had been worn during the late [[Regency period]], had high, small crowns and brims that grew larger until the 1830s, when the face of a woman wearing a poke bonnet could only be seen directly from the front. They had rounded brims, echoing the rounded form of the bell-shaped hoop skirts.

Bonnets shrunk at the end of the 1860s and moved to a perched position in the early 1870s as hairstyles grew in scale and intricacy. This led to the popularization of hats, which became the headwear of choice for the remainder of the Victorian era.&lt;ref name=&quot;g4223&quot;&gt;{{cite book |last=Cunnington |first=Cecil Willett |title=English Women's Clothing in the Nineteenth Century |date=1990-05-01 |publisher=Courier Corporation |isbn=0-486-26323-1 |publication-place=New York |page=}}&lt;/ref&gt;
[[File:The_London_and_Paris_ladies'_magazine_(Apr_1885)_03.png|thumb|Flower pot style hat of 1885.]]
The 1880s saw a hat inspired by the top hat for women known as the flowerpot hat, and the 1890s saw the popularity of the boater. The hats of the late Victorian era were covered with elaborate creations of silk flowers, ribbons, and above all, exotic plumes; hats sometimes included entire exotic birds that had been stuffed. Many of these plumes came from birds in the Florida everglades, which were nearly made entirely extinct by overhunting. By 1899, early environmentalists like [[Adeline Knapp]] were engaged in efforts to curtail the hunting for plumes. By 1900, more than five million birds a year were being slaughtered, and nearly 95 per cent of Florida's shore birds had been killed by [[Plume hunting|plume hunter]]s.&lt;ref&gt;{{cite web|title=Everglades National Park|url=https://www.pbs.org/nationalparks/parks/everglades/|archive-url=https://web.archive.org/web/20090927085907/http://www.pbs.org/nationalparks/parks/everglades/|url-status=dead|archive-date=27 September 2009|publisher=PBS|access-date=7 November 2011}}&lt;/ref&gt;

== Shoes ==
The women's shoes of the early Victorian period were narrow and heelless, in black or white satin. By 1850s and 1860s, they were slightly broader with a low heel and made of leather or cloth. Ankle-length laced or buttoned boots were also popular. From the 1870s to the twentieth century, heels grew higher and toes more pointed. Low-cut pumps were worn for the evening.&lt;ref name=&quot;:4&quot; /&gt;

== Cosmetics ==
[[Victorian-era cosmetics]] were typically less obvious than ours. However, small amounts of pale face powder or powdered blush were widely used.&lt;ref&gt;{{Cite book |last=Goodman |first=Ruth |title=How to be a Victorian |date=2014 |publisher=Penguin Books |isbn=978-0-670-92136-2 |location=London}}&lt;/ref&gt; Tints were sometimes added to face creams. Some cosmetics contained toxic or caustic ingredients like lead, mercury, ammonia, and arsenic {{Citation needed|date=October 2025}}.

Hair color
==Mourning black==
{{See also |Mourning stationery}}
[[File:The royal children in mourning Mar 1862.jpg|thumb|Victoria's five daughters (Alice, Helena, Beatrice, Victoria and Louise), photographed wearing mourning black beneath a bust of their late father, Prince Albert (1862)]]
[[File:Mourning dress MET 50.40.3a-b front CP4.jpg|alt=Black Victorian mourning dress|thumb|Mourning Dress, 1894–95]]

In Britain, black is the colour traditionally associated with mourning for the dead. The customs and etiquette expected of men, and especially women, were rigid but evolving during much of the Victorian era. The expectations depended on a complex hierarchy of close or distant relationship with the deceased. (Davidoff) The closer the relationship, the longer the mourning period and the wearing of black. The wearing of full black was known as First Mourning, which had its own expected attire, including fabrics, and an expected duration of 4 to 18 months. Following the initial period of First Mourning, the mourner would progress to Second Mourning, a transition period of wearing less black, which was followed by Ordinary Mourning, and then Half-mourning. Some of these stages of mourning were shortened or skipped completely if the mourner's relationship to the deceased was more distant. Half-mourning was a transition period when black was replaced by acceptable colours such as lavender and mauve, possibly considered acceptable transition colours because of the tradition of [[Church of England]] (and [[Catholic Church|Catholic]]) clergy wearing lavender or mauve [[Stole (vestment)|stoles]] for funeral services, to represent the [[Passion (Christianity)|Passion of Christ]].&lt;ref&gt;{{cite web|title=The Colors of the Church Year|url=http://fullhomelydivinity.org/articles/colors.htm|publisher=Consortium of Country Churches|access-date=6 November 2011|archive-date=13 November 2011|archive-url=https://web.archive.org/web/20111113075214/http://fullhomelydivinity.org/articles/colors.htm|url-status=dead}}&lt;/ref&gt;

The mourning dress worn by Queen Victoria (below, right) &quot;shows the traditional touches of mourning attire, which she wore from the death of her husband, Prince Albert (1819–1861), until her own death.&quot;&lt;ref&gt;{{Cite web|url=https://www.metmuseum.org/art/collection/search/155839?&amp;searchField=All&amp;sortBy=Relevance&amp;deptids=8&amp;ft=queen+victoria&amp;offset=0&amp;rpp=20&amp;amp;pos=2|title=Mourning Dress, 1894–95|last=The Metropolitan Museum of Art|date=7 September 2019|website=The Metropolitan Museum of Art|access-date=7 September 2019}}&lt;/ref&gt; Dating from 1894–95, Queen Victoria wore this dress as a result of the death of the eldest son of the Prince and Princess of Wales, Eddy, in line to the throne.

=== Norms for mourning===

''Manners and Rules of Good Society, or, Solecisms to be Avoided'' (London, Frederick Warne &amp; Co., 1887) gives clear instructions, such as the following:&lt;ref&gt;{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–83}}&lt;/ref&gt;

{| class=&quot;wikitable&quot;
|-
! Relationship to deceased !! First mourning !! Second mourning !! Ordinary mourning !! Half-mourning
|-
| Wife for husband || 1-year, 1-month; [[bombazine]] fabric covered with [[Crape|crepe]]; [[widow's cap]], [[lawn cuff]]s, collars || 6 months: less crepe || 6 months: no crepe, silk or wool replaces bombazine; in last 3 months jet jewellery and ribbons can be added || 6 months: colours permitted are grey, lavender, mauve, and black-and-grey
|-
| Daughter for parent || 6 months: black with black or white crepe (for young girls); no linen cuffs and collars; no jewellery for first 2 months || 4 months: less crepe ||  –  || 2 months as above
|-
| Wife for husband's parents || 18 months in black bombazine with crepe || – || 3 months in black || 3 months as above
|-
| Parent for son- or daughter-in-law's parent || – Black armband in representation of someone lost || – || 1-month black || –
|-
| Second wife for parent of a first wife || – || – || 3 months black || –
|}

The complexity of these etiquette rules extends to specific mourning periods and attire for siblings, step-parents, aunts and uncles distinguished by blood and by marriage, nieces, nephews, first and second cousins, children, infants, and &quot;connections&quot; (who were entitled to ordinary mourning for a period of &quot;1–3 weeks, depending on level of intimacy&quot;). Men were expected to wear mourning black to a lesser extent than women, and for a shorter mourning period. After the mid-19th century, men would wear a black hatband and black suit, but for only half the prescribed period of mourning expected of women. Widowers were expected to mourn for a mere three months, whereas the proper mourning period expected for widows was up to four years.&lt;ref&gt;{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–9}}&lt;/ref&gt; Women who mourned in black for longer periods were accorded great respect in public for their devotion to the departed, the most prominent example being Queen Victoria herself. It was not uncommon for a widow who did not remarry to wear half-mourning for the rest of her life, except when another death necessitated full mourning. For example, Alexandra, Princess of Wales wore half-mourning for the rest of her life after her eldest son Eddy died in 1894. Empress Elisabeth of Austria did the same, as did Empress Eugénie of France. They reverted to full mourning when appropriate, but they never wore less than half-mourning after their sons' deaths.

Women with lesser financial means tried to keep up with the example being set by the middle and upper classes by dyeing their daily dress. Dyers made most of their income during the Victorian period by dyeing clothes black for mourning.&lt;ref&gt;{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|page=341}}&lt;/ref&gt;

== Technological advancement ==
The technological changes that affected the manufacture and consumption of clothing in the Victorian age included the following:

* the mass production of fabrics — for example, &quot;by the early 1850s there were thousands of steam-powered looms churning out millions of miles of fabric every year&quot;  [62]
* the invention of aniline dyes, which were much more vibrantly colored and resistant to fading than the natural dyes that had been used. — . Invented by chemist [[William Henry Perkin]] in 1856, the first aniline dye mauveine (or mauve) &quot;wash[ed] the fashionable landscape in a haze of purple.&quot;&lt;ref name=&quot;:22&quot;&gt;{{Cite book|title=The Dress Diary: Secrets from a Victorian Woman's Wardrobe|last=Strasdin|first=Kate|publisher=Pegasus Books|year=2023|location=New York, New York}}&lt;/ref&gt; (247) Other intense and, to the Victorians, intensely exciting colors followed, but the new synthetic additions to fabric sometimes included chemicals harmful to their wearers. For example, a &quot;bright-magenta hue was achieved by adding arsenical-based chemicals to existing aniline dyes, brightening the already luminous shades – but these left residues themselves, along with a toxic labour trail in their wake.&quot;&lt;ref name=&quot;:22&quot; /&gt; (255) Perhaps the most famous of these is arsenic green, used on fabrics, wallpapers, and trim: &quot;The craze for artificial foliage to adorn the heads and dresses of women of fashion in the mid-nineteenth century had seen the proliferation of flower workshops, where young women in their hundreds laboured to produce the lifelike green leaves and blooms that would make a fetching headdress or would trail becomingly across the bodice of a gown. The lushness of the green was achieved by the application of a powder, a pigment that was created by mixing copper and the highly toxic chemical, arsenic trioxide. The physical effects of working with this poisonous compound were horrific. Contemporary medical drawings depict the green hue of the skin and dreadful open lesions on the hands of the maker, whilst the daily gradual ingestion of the powder by the flower girls was eventually fatal.&quot;&lt;ref name=&quot;:22&quot; /&gt; (255)
* the invention of a sewing machine that could be used in the home. Although sewing machines were already in use in the clothing industry, in 1858 Isaac Merritt Singer began to sell &quot;lightweight domestic machines&quot; for home sewing, radically increasing women's control over their own dress.&lt;ref name=&quot;:24&quot;&gt;{{Cite book|title=Victorian Fashions for Women|last=Kay|first=Fiona|last2=Storey|first2=Neil R.|publisher=Pen &amp; Sword History|year=2022|isbn=978 1 39900 416 9|location=Yorkshire and Philadelphia|pages=}}&lt;/ref&gt; (91 [of 298])
* the spread of journalism for women and fashion journalism
* the Jacquard loom: &quot;French inventor Joseph-Marie Jacquard’s loom attachment mechanized the weaving of figured fabrics by 1804.&quot;&lt;ref name=&quot;:25&quot; /&gt; (454)

Perhaps not at the same scale as these but as important in 1850s designs was a technology that turned iron into steel, which could then be drawn into fine wires.&lt;ref name=&quot;:3&quot;&gt;{{Cite book|title=The Culture of Fashion|last=Breward|first=Christopher|publisher=Manchester University Press|year=1995|pages=145–180}}&lt;/ref&gt; Steel was refined to a malleable state so that thin blades could be curved into concentric circles (called hoops) and connected with wires to form the cage.

Technological advancements not only influenced the economy but brought a major change in the fashion styles worn by men and women. As the Victorian era was based on the principles of gender, race and class.&lt;ref&gt;{{cite journal|last1=Graham|first1=P|title=The Victorian Era|url=https://archive.org/details/in.ernet.dli.2015.261548|journal=Digital Library of India}}&lt;/ref&gt; Much advancement was in favor of the upper class as they were the ones who could afford the latest technology and change their fashion styles accordingly.  In 1830s there was introduction of horse hair crinoline that became a symbol of status and wealth as only the upper-class women could wear it.  In 1850s there were more fashion technological advancements hence 1850s could rightly be called a revolution in the Victorian fashion industry such as the innovation of artificial cage crinoline that gave women an artificial hourglass silhouette without layers of petticoats, which was lighter and more hygienic.&lt;ref&gt;{{cite book|last1=Shrimpton|first1=J|title=Victorian Fashion|publisher=Bloomsbury Shire Publications}}&lt;/ref&gt; Synthetic dyes, such as [[mauveine]] (aniline purple), were introduced in 1856, adding bright colours to garments. In 1855's ''[[Haute couture]]'' was introduced as tailoring became more mainstream in years to follow.&lt;ref&gt;{{cite book|last1=Aspelund|first1=Karl|title=Fashioning Society|publisher=Fairchild Books}}&lt;/ref&gt;

Charles Frederick Worth, a prominent English designer, became popular amongst the upper class though its city of destiny always is Paris. Haute couture became popular at the same time that sewing machines were invented.&lt;ref name=&quot;Haute Couture&quot;&gt;{{cite book|last1=Martin|first1=Richard|last2=Koda|first2=Harold|title=Haute Couture|publisher=The Metropolitan Museum of Art}}&lt;/ref&gt; Princess [[Eugénie de Montijo|Eugenie]] of France wore the Englishman dressmaker, Charles Frederick Worth's couture and he instantly became famous in France though he had just arrived in Paris a few years ago. In 1855, Queen Victoria and Prince Albert of Britain welcomed [[Napoleon III]] and Eugenie of France to a full state visit to England. Eugenie was considered a fashion icon in France. Queen Victoria, who had been the fashion icon for European high fashion, was inspired by Eugenie's style and the fashions she wore.{{Citation needed|date=October 2025}} Later, Queen Victoria also appointed Charles Frederick Worth as her dress maker and he became a prominent designer amongst the European upper class. Charles Frederick Worth is known as the father of the haute couture as later the concept of labels were also invented in the late 19th century as custom, made to fit tailoring became mainstream.&lt;ref&gt;{{cite book|last1=Saillard|first1=Olivier|last2=Zazzo|first2=Anne|title=Paris Haute Couture|publisher=Skira Flammarion}}&lt;/ref&gt;

By the 1860s, when made-to-fit tailoring was popular in Europe, crinolines were considered impractical. In the 1870s, women preferred more slimmer silhouettes, hence bodices grew longer and the polonaise, a skirt and bodice made together, was introduced. In 1870s the Cuirass Bodice, a piece of armour that covers the torso and functions like a corset, was invented. Towards the end of Victoria's reign, dresses were flared naturally as crinolines were rejected by middle-class women. Designers such as Charles Frederick Worth were also against them. All these inventions and changes in fashion led to women's liberation as tailored looks improved posture and were more practical.&lt;ref name=&quot;Haute Couture&quot;/&gt;

dressmakers, couturiers, modistes

== Home decor ==
{{main|Victorian decorative arts}}

Home decor started spare, veered into the elaborately draped and decorated style we today regard as Victorian, then embraced the retro-chic of [[William Morris]] as well as pseudo-[[Japonaiserie]].

== Myths and Oversimplifications ==

=== Modesty ===
{{main|Victorian morality}}
{{Original research|section|date=May 2008}}
[[File:1868-skirt-lengths-girl-ages-Harpers-Bazar.gif|thumb|upright|&quot;The proper length for little girls' skirts at various ages&quot;, from ''[[Harper's Bazaar]]'', showing a 1900 idea of how the hemline should descend towards the ankle as a girl got older]]Many myths and exaggerations about the period persist to the modern day. Examples include the idea of men's clothing is seen as formal and stiff, women's as elaborate and over-done; clothing covered the entire body, and even the glimpse of an ankle was scandalous. Critics contend that [[corset]]s constricted women's bodies and lives. Homes are described as gloomy, dark, cluttered with massive and over-ornate furniture and proliferating [[bric-a-brac]]. Myth has it that even piano legs were scandalous, and covered with tiny [[pantalette]]s.

=== Tight Lacing ===
Tight-lacing, which was not possible until the development of the grommet in 1828, was famously controversial in the Victorian age, generating many column inches of profitable newspaper copy, in part because it was (and still is) fetishistic and subversive in that adolescent girls used it as a means of rebellion and upper-working- or lower-middle-class shop girls saw it as a means of upward mobility.&lt;ref name=&quot;:21&quot;&gt;{{Cite book|title=Fashion and Fetishism: Corsets, Tight-Lacing and Other Forms of Body-sculpture|last=Kunzle|first=David|publisher=History Press|year=2013|isbn=978 0 7524 9545 3|location=Stroud, Gloucestershire|pages=}}&lt;/ref&gt; (71 [of 1182]) No evidence exists that tight lacing was widespread or particularly dangerous.&lt;ref name=&quot;:21&quot; /&gt; ()


In truth, men's formal clothing may have been less colourful than it was in the previous century, but brilliant [[waistcoat]]s and [[cummerbund]]s provided a touch of colour, and [[smoking jacket]]s and [[robe|dressing gown]]s were often of rich Oriental [[brocade]]s. This phenomenon was the result of the growing textile manufacturing sector, developing mass production processes, and increasing attempts to market fashion to men.&lt;ref name=&quot;shannon597&quot;/&gt; Corsets stressed a woman's sexuality, exaggerating hips and bust by contrast with a tiny waist. Women's [[evening gown]]s bared the shoulders and the tops of the breasts. The [[jersey dress]]es of the 1880s may have covered the body, but the stretchy novel fabric fit the body like a glove.&lt;ref&gt;{{cite book |last=Gernsheim |first=Alison |title=Victorian &amp; Edwardian Fashion: A Photographic Survey |year=1981 |publisher=Dover Publications |location=New York |page=65|edition=New |isbn=0-486-24205-6}}&lt;/ref&gt;

Home furnishing was not necessarily ornate or overstuffed. However, those who could afford lavish draperies and expensive ornaments, and wanted to display their wealth, would often do so. Since the Victorian era was one of increased social mobility, there were ever more ''[[nouveaux riches]]'' making a rich show.

The items used in decoration may also have been darker and heavier than those used today, simply as a matter of practicality. London was noisy and its air was full of [[soot]] from countless coal fires. Hence those who could afford it draped their windows in heavy, sound-muffling curtains, and chose colours that didn't show soot quickly. When all washing was done by hand, curtains were not washed as frequently as they might be today.

There is no actual evidence that piano legs were considered scandalous. Pianos and tables were often draped with [[shawl]]s or cloths—but if the shawls hid anything, it was the cheapness of the furniture. There are references to lower-middle-class families covering up their [[pine]] tables rather than show that they couldn't afford [[mahogany]]. The piano leg story seems to have originated in the 1839 book, ''A Diary in America'' written by Captain [[Frederick Marryat]], as a satirical comment on American prissiness.&lt;ref&gt;{{cite book |last1=Marryat |first1=C.B. |title=A Diary in America: With Remarks on Its Institutions |date=1839 |publisher=Longman, Orme, Brown, Green, and Longmans |location=London, England |volume=2 |pages=246–247 |url=https://books.google.com/books?id=2-VEAAAAIAAJ&amp;pg=PA246}} From pp. 246-247:  &quot;I was requested by a lady to escort her to a seminary for young ladies, and on being ushered into the reception-room, conceive my astonishment at beholding a square piano-forte with four ''limbs''. However, that the ladies who visited their daughters, might feel in its full force the extreme delicacy of the mistress of the establishment, and her care to preserve in their utmost purity the ideas of the young ladies under her charge, she had dressed all these four limbs in modest little trousers, with frills at the bottom of them!&quot;&lt;/ref&gt;

Victorian manners may have been as strict as imagined—on the surface. One simply did not speak publicly about sex, childbirth, and such matters, at least in the respectable middle and upper classes. However, as is well known, discretion covered a multitude of sins. Prostitution flourished. Upper-class men and women indulged in [[adultery|adulterous]] liaisons.

== Gallery ==
{{gallery
|2=A mid-Victorian interior: ''Hide and Seek'' by [[James Tissot]], c. 1877
Image:Winterhalter Elisabeth.jpg|3=Dress designed by [[Charles Frederick Worth]] for [[Elisabeth of Bavaria|Elisabeth of Austria]] painted by [[Franz Xaver Winterhalter]].|4=File:Frith A Private View detail.jpg|5=[[William Powell Frith]]'s painting of 1883 contrasts women's [[Aesthetic dress]] (left and right) with fashionable attire (center).|6=File:Tissot lilacs 1875.jpg|7=Day dress, c. 1875 [[James Tissot]] painting.|8=File:James Abbot McNeill Whistler 011.jpg|9=[[James McNeill Whistler|Whistler]]'s [[Portrait of Lady Meux]], 1882
Image:Jeanna_Samary-Renoir.png|10=[[Pierre-Auguste Renoir|Renoir]]'s portrait of [[Jeanne Samary]] in an [[evening gown]], 1878|11=File:Melville_-_Queen_Victoria.jpg|12=Portrait by [[Alexander Melville (artist)|Alexander Melville]] of [[Victoria of the United Kingdom|Queen Victoria]], 1845|13=File:Henry Treffry Dunn Rossetti and Dunton at 16 Cheyne Walk.jpg|14=An artistic interior: [[Dante Gabriel Rossetti]] reading to [[Theodore Watts-Dunton]] in the drawing room at No. 16 [[Cheyne Walk]], 1882|15=File:Punch - Masculine beauty retouched1.png|16=Men's swimwear: Cartoon from ''[[Punch (magazine)|Punch]]'' by [[George du Maurier]]}}

== See also ==
* [[Emily Clapham]]
* [[Victorian decorative arts]]
* [[Victorian dress reform]]
* [[Victorian morality]]
* [[Victoriana]]
* [[Women in the Victorian Era]]
* [[Charles Frederick Worth]]

=== Time periods ===
* [[1830s in fashion]]
* [[1840s in fashion]]
* [[1850s in fashion]]
* [[1860s in fashion]]
* [[1870s in fashion]]
* [[1880s in fashion]]
* [[1890s in fashion]]

=== Women's clothing ===
* [[Corset]]
* [[Corset controversy]]
* [[Tightlacing]]
* [[Bloomers (clothing)|Bloomers]]
* [[Bodice]]

=== Contemporary interpretations ===
* [[Steampunk]]
* [[Neo-Victorian]]
* [[Lolita Fashion|Lolita]]

== References ==
{{Reflist}}

== Further reading ==
*{{cite book |author=Phipps, Elena| title= ''From Queen to Empress: Victorian dress 1837-1877'' | location=New York | publisher=The Metropolitan Museum of Art | year=1988 | isbn=0870995340| url= http://libmma.contentdm.oclc.org/cdm/compoundobject/collection/p15324coll10/id/69547/rec/235 | display-authors=etal}}
* Sweet, Matthew – ''Inventing the Victorians'', St. Martin's Press, 2001 {{ISBN|0-312-28326-1}}

== External links ==
* [http://www.victorians.co.uk/victorian-fashion Victorian Fashion] {{Webarchive|url=https://web.archive.org/web/20180407223711/http://www.victorians.co.uk/victorian-fashion |date=7 April 2018 }}
* [https://www.victorianvoices.net/topics/fashion/index.shtml VictorianVoices.net] – Fashion articles and illustrations from Victorian periodicals; extensive fashion image gallery
* [http://www.cracked.com/article_19575_5-ridiculous-sex-myths-from-history-you-probably-believe.html Victorian myths]
* [http://www.victorianstation.com/lifestylemenu.htm Victorian fashion, etiquette, and sports] {{Webarchive|url=https://web.archive.org/web/20180103162620/http://www.victorianstation.com/lifestylemenu.htm |date=3 January 2018 }}
* [http://www.thesmartset.com/article/article12180701.aspx Background on &quot;A Diary in America&quot;]
* [http://www.mccord-museum.qc.ca/en/keys/webtours/VQ_P2_17_EN.html Form and Fashion] —  the evolution of women's dress during the 19th century (many photographs)
* [http://www.mccord-museum.qc.ca/en/keys/games/jeu2/ Educational Game: Mix and Match] — build a 19th-century dress using a virtual mannequin
* {{cite web |publisher= [[Victoria and Albert Museum]]
 |url= http://www.vam.ac.uk/content/articles/v/victorian-dress-at-v-and-a/
 |title= Victorian Dress
 |work= Fashion, Jewellery &amp; Accessories
 |date= 14 January 2011
 |access-date= 2011-04-03}}
*[http://cv.vic.gov.au/stories/creative-life/fashion-detective-fashion-fiction-and-forensics/ Fashion detective: Fashion, Fiction and Forensics in nineteenth century Australian fashion] on Culture Victoria

{{Timeline of clothing and fashion|state=collapsed}}{{Victorian era|state=collapsed}}

[[Category:Victorian fashion| ]]
[[Category:19th-century fashion|*]]
[[Category:1900s fashion]]
[[Category:History of Western fashion]]
[[Category:19th century in the arts]]

=From ''Women in the Victorian era''=
===Victorian women's fashion===
{{Multiple issues|{{tone|date=March 2023}}
{{more footnotes needed|date=March 2023}}|section=y}}{{Further|Victorian fashion}}
The ideal Victorian woman was pure, chaste, refined, and modest. This ideal was supported by etiquette and manners. The etiquette extended to the pretension of never acknowledging the use of undergarments (sometimes generically referred to as &quot;unmentionables&quot;). The discussion of such a topic, it was feared, would gravitate towards unhealthy attention on anatomical details. As one Victorian lady expressed it: &quot;[those] are not things, my dear, that we speak of; indeed, we try not even to think of them&quot;, in contrast to current norms.&lt;ref&gt;{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=20}}&lt;/ref&gt; The pretence of avoiding acknowledgement of anatomical realities met with embarrassing failure on occasion. In 1859, the Hon. Eleanor Stanley wrote about an incident where the [[Louisa Cavendish, Duchess of Devonshire|Duchess of Manchester]] moved too quickly while manoeuvring over a [[stile]], tripping over her large [[hoop skirt]]:

{{blockquote|[the Duchess] caught a hoop of her cage in it and went regularly head over heels lighting on her feet with her cage and whole petticoats above, above her head. They say there was never such a thing seen – and the other ladies hardly knew whether to be thankful or not that a part of her undergarments consisted in a pair of scarlet tartan [[knickerbockers (clothing)|knickerbockers]] (the things Charlie shoots in) which were revealed to the view of all the world in general and the [[Aimable Pélissier|Duc de Malakoff]] in particular&quot;.&lt;ref&gt;{{cite book|last=Cunnington|first=C. Willett|title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations|year=1990|publisher=Dover Publications|isbn=978-0-486-26323-6|pages=20–1}}&lt;/ref&gt;}}

However, despite the fact that Victorians considered the mention of women's undergarments in mixed company unacceptable, men's entertainment made great comedic material out of the topic of ladies' [[bloomers (clothing)|bloomers]], including men's magazines and music hall skits.&lt;ref&gt;{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=22}}&lt;/ref&gt;

Victorian women's clothing followed trends that emphasised elaborate dresses, skirts with wide volume created by the use of layered material such as [[crinoline]]s, hoop skirt frames, and heavy fabrics. Because of the impracticality and health impact of the era's fashions, a [[Victorian dress reform|dress reform movement]] began among women.

The ideal silhouette of the time demanded a narrow waist, which was accomplished by constricting the abdomen with a laced [[corset]]. While the silhouette was striking, and the dresses themselves were often exquisitely detailed creations, the fashions were cumbersome. At best, they restricted women's movements and at worst, they had a harmful effect on women's health. Physicians turned their attention to the use of corsets and determined that they caused several medical problems: compression of the thorax, restricted breathing, organ displacement, poor circulation, and prolapsed uterus.&lt;ref name=&quot;O'Connor&quot;/&gt;

Articles advocating the reform of women's clothing by the British National Health Society, the Ladies' Dress Association, and the [[Rational Dress Society]] were reprinted in ''The Canada Lancet'', Canada's medical journal. In 1884, Dr J. Algernon Temple of Toronto even voiced concern that the fashions were having a negative impact on the health of young women from the working classes. He pointed out that a young working-class woman was likely to spend a large part of her earnings on fine hats and shawls, while &quot;her feet are improperly protected, and she wears no flannel petticoat or woollen stockings&quot;.&lt;ref name=&quot;O'Connor&quot;/&gt;
[[File:Bloomers.jpg|thumb|1850s illustration of a woman wearing [[bloomers]]]]
[[Florence Pomeroy]], Lady Haberton, was president of the Rational Dress movement in Britain. At a National Health Society exhibition held in 1882, Viscountess Haliburton presented her invention of a &quot;[[divided skirt]]&quot;, which was a long skirt that cleared the ground, with separate halves at the bottom made with material attached to the bottom of the skirt. She hoped that her invention would become popular by supporting women's freedom of physical movement, but the British public was not impressed by the invention, perhaps because of the negative &quot;unwomanly&quot; association of the style with the American [[Bloomers]] movement.&lt;ref&gt;{{cite book|last=Murray|first=Janet Horowitz|title=Strong-Minded Women and Other Lost Voices from 19th Century England|year=1982|publisher=Pantheon Books|location=New York|isbn=0-394-71044-4|pages=[https://archive.org/details/strongmindedwome00jane/page/68 68–70]|url=https://archive.org/details/strongmindedwome00jane/page/68}}&lt;/ref&gt; [[Amelia Jenks Bloomer]] had encouraged the wearing of visible bloomers by feminists to assert their right to wear comfortable and practical clothing, but it was no more than a passing fashion itself among radical feminists. The movement to reform women's dress would persist and have long-term success, however; by the 1920s, [[Coco Chanel]] was successful at selling a progressive, far less restrictive silhouette that abandoned the corset and raised hemlines. The new silhouette symbolised modernism for trendy young women and became the 20th century standard. Other Paris designers continued reintroducing pants for women and the trend was gradually adopted over the next century.

Fashion trends, in one sense, travelled &quot;full circle&quot; over the course of the Victorian era. The popular women's styles during the [[Georgian era]], and at the very beginning of Victoria's reign, emphasized a simple style influenced by flowing gowns worn by women in [[Ancient Greek clothing|Ancient Greece]] and [[Clothing in ancient Rome|Rome]]. The [[Empire waist]] silhouette was replaced by a trend towards ornate styles and an artificial silhouette, with the restrictiveness of women's clothing reaching its low point during the mid-century passion for narrow corseted waists and hoop skirts. The iconic wide-brimmed women's hats of the later Victorian era also followed the trend towards ostentatious display. Hats began the Victorian era as simple [[Bonnet (headgear)|bonnets]]. By the 1880s, milliners were tested by the competition among women to top their outfits with the most creative (and extravagant) hats, designed with expensive materials such as silk flowers and exotic plumes such as ostrich and peacock. As the Victorian era drew to a close, however, fashions were showing indications of a popular backlash against excessive styles. Model, actress and socialite [[Lillie Langtry]] took London by storm in the 1870s, attracting notice for wearing simple black dresses to social events. Combined with her natural beauty, the style appeared dramatic. Fashions followed her example (as well as Queen Victoria's wearing of mourning black later in her reign). According to [[Harold Koda]], the former Curator-in-chief of the [[Costume Institute at The Met|Metropolitan Museum of Art's Costume Institute]],&lt;ref&gt;{{cite web|url=http://www.metmuseum.org/about-the-museum/press-room/exhibitions/2014/death-becomes-her|title=Death Becomes Her: A Century of Mourning Attire : October 21, 2014-February 1, 2015|website=Metmuseuim.org|access-date=7 November 2021}}&lt;/ref&gt; &quot;The predominantly black palette of [[mourning]] dramatizes the evolution of period silhouettes and the increasing absorption of fashion ideals into this most codified of etiquettes,&quot; said Koda, &quot;The veiled widow could elicit sympathy as well as predatory male advances. As a woman of sexual experience without marital constraints, she was often imagined as a potential threat to the social order.&quot;

====Evolution of Victorian women's fashion====
&lt;gallery&gt;
File:Fashion plate December 1844.jpg|Ladies' December Fashions (1844). Hand-coloured steel engraving from a women's magazine.
File:Thegalleryofhmscalcutta james tissot 1876.jpg|''[[The Gallery of HMS Calcutta]]'' by [[James Tissot]] (1876). [[Bustle]]s were fashionable in the 1870s and 1880s.
File:Mrs lillie langtry george frederic watts 1880.jpg|''Mrs. Lillie Langtry'' by [[George Frederic Watts]] (1880).
File:Five-women-on-queenslander-steps-r.jpg|Fashionable women in [[Queensland]], Australia around 1900.
&lt;/gallery&gt;


{{Short description|Irish writer (born 1963)}}
{{Use Irish English|date=August 2025}}
{{Use dmy dates|date=August 2025}}
{{Infobox writer
| name               = Darach Ó Scolaí
| image              = Darach Ó Scolaí.JPG
| alt                = Man holding prize-winning book
| caption            = Ó Scolaí in 2019
| birth_name         = Darach Ó Scolaí
| birth_date         = {{Birth date and age|1963|df=y}}
| birth_place        = [[County Galway]], The Republic of Ireland
| death_date         = 
| death_place        = 
| occupation         = Writer, artist, publisher
| alma_mater         = [[University of Galway]]
| years_active       = 1998–present
| genre            = Novel, retelling, translation, play, screenplay, illustrated book for children and adults
| other_names        = 
| spouse             = 
| children           = 3
| awards             = [[Awards and Honors received by Darach Ó Scolaí|Awards and Honors]]
| signature          = 
| website            = 
}}[[File:Darach Ó Scolaí.JPG|thumb|Darach Ó Scolaí, holding ''Oileán an Órchiste'' (his translation of Robert Louis Stevenson's ''Treasure Island'')]]


== Darach Ó Scolaí ==
Darach Ó Scolaí (&lt;small&gt;Irish:&lt;/small&gt; [/ˈda.rax/ /oː/ /sˠkˠoː/l̪ˠəi/]; born 1963&lt;ref&gt;{{Cite web|url=https://portraidi.ie/en/darach-o-scolai/|title=Darach Ó Scolaí|date=20 October 2017|website=Portráidí (Portraits of Irish-Language Writers)|access-date=1 August 2025}}&lt;/ref&gt;) is an Irish author who works in a number of genres, from novels, plays and screenplays to illustrated books for children and adults. He began his literary career in 1998 writing screenplays, stage plays, retellings and translations; he began to publish novels in 2008. Ó Scolaí is widely recognized as a leading figure in contemporary Irish literature, known as “one of the most important Irish language writers of his generation”&lt;ref&gt;{{Cite journal|last=Poirtéir|first=Cathal|date=2022|title=? Suil an Daill: Constant Tensions and Shifting Allegiances|url=https://booksirelandmagazine.com/suil-an-daill-constant-tensions-and-shifting-allegiances/|journal=Books Ireland}}&lt;/ref&gt; and &quot;one of the great Irish language novelists [duine d’úrscéalaithe móra na Gaeilge].&quot;&lt;ref name=&quot;:17&quot; /&gt; His writing has been called “the high literature of the Irish language.”&lt;ref&gt;Ó Coimín, Maitiú. ''Nós'' 2 February 2018). Qtd. in &quot;Táin Bó Cuailnge.&quot; ''Leabhar Breac''. Retrieved 25 August 2025.&lt;/ref&gt;

Much of his fiction is based on a knowledge of traditional Irish tales and narrative practices as well as Irish history. He specializes in literary and [[wikipedia:Historical_fiction|historical fiction]], or as novelist Alan Titley says, Ó Scolaí’s “peak (for now), or at least his greatest imaginative interest, is the historical novel [tá an chuma air gurb é a bhuaic (go fóill), nó ar a laghad, a mhórspéis samhlaíochta, an t-úrscéal staire].”&lt;ref name=&quot;:11&quot;&gt;{{Cite journal|last=Titley|first=Alan|date=Fall 2020|title=An Stíl Go Deo!: Soather Dharach Uí Scolaí (The style would be forever!: Worker Darach Ó Scolaí)|url=https://www.jstor.org/stable/27046090|journal=Comhar|volume=80, No. 10|pages=27|via=JSTOR}}&lt;/ref&gt; His retellings of old stories and tales from their original Middle and Early-Modern Irish into Modern Irish ([[wikipedia:Irish_language|Gaeilge]]) are respected for their accessibility to students and language learners as well as for their artistry.

Ó Scolaí also regularly reviews books and lectures and writes on literature and culture. 

Beyond his writing, Ó Scolaí is a publisher and has co-produced a number of film, television shows and stage plays. 

== Life ==
Ó Scolaí was born in Dublin and raised in the Galway [[wikipedia:Gaeltacht#Galway Gaeltacht|Gaeltacht]] (Irish-speaking) regions of Cois Fharraige on the north shore of Galway Bay, in the Republic of Ireland, where he lives now with his wife and children in Lochán Beag (Indreabhán).&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|date=30 October 2024|title=Duais don úrscéal liteartha is fearr buaite ag Darach Ó Scolaí ag Oireachtas na Samhna|url=https://tuairisc.ie/duais-don-ursceal-liteartha-is-fearr-buaite-ag-darach-o-scolai-ag-oireachtas-na-samhna/|journal=Tuairisc}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Ní Scolaí|first=Aifric|date=2024|title=Darach Ó Scolaí|url=https://www.taiscecf.ie/ealaiontoiri?category=Scr%C3%ADbhneoir|journal=Taisce Chois Fharraige}}&lt;/ref&gt; 

He graduated the [[wikipedia:University_of_Galway|University of Galway]] (then University College Galway) with a B.A. in 1983.&lt;ref&gt;{{Cite web|url=https://www.linkedin.com/in/darach-ó-scolaí-20026920/|title=Darach Ó Scolaí|last=Ó Scolaí|first=Darach|date=August 2025|website=LinkedIn}}&lt;/ref&gt; 

=== Writing and Publishing ===
Ó Scolaí writes in Irish ([[wikipedia:Irish_language|Gaeilge]]), his native language, and lives in an area defined for the predominant presence of Irish as the vernacular language, the language spoken at home. Irish was the language of his parents' home and is the language of children as well. He is fluent in Irish and English and conversant in French. 

None of his works has been translated into English. 

==== Leabhar Breac ====
In 1995 Darach Ó Scolaí and his brother Caomhán Ó Scolaí — a [[wikipedia:Typography|typographer]] and designer — founded the publishing house Leabhar Breac at Indreabhán (Inverin), County Galway. Their father “Séamas Ó Scolaí was an editor at An Gúm and worked on the Irish-English dictionary team [bhí a n-athair Séamas Ó Scolaí ina eagarthóir sa Ghúm agus d’oibrigh sé ar fhoireann an fhoclóra Gaeilge-Béarla].”&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/about-us/|title=About Us|date=2024|website=Leabhar Breac|access-date=1 July 2025}}&lt;/ref&gt; Darach Ó Scolaí has been publisher and literary editor at Leabhar Breac since its founding. 

Named for [[wikipedia:An_Leabhar_Breac|An Leabhar Breac (The Speckled Book)]], Leabhar Breac publishing house has more than 140 books in print.&lt;ref name=&quot;:0&quot; /&gt; Leabhar Breac aims to publish Irish-language books that meet “a high literary and artistic standard.”&lt;ref name=&quot;:0&quot; /&gt; Besides the content, Leabhar Breac is known for the typically &quot;superb [thar cionn]&quot; quality of the design and production of the &quot;physical book [leabhar fisiciúil].&quot;&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Ní Mhuilneoir|first=Gráinne|date=30 July 2024|title=‘Bláthnaid’ – leabhar álainn i sraithín álainn faoi mhná|url=https://tuairisc.ie/blathnaid-leabhar-alainn-i-sraithin-alainn-faoi-mhna/|journal=Tuairisc}}&lt;/ref&gt; Its books regularly win awards for literary and artistic quality. Leabhar Breac also publishes translations for children and adults from various early versions of Irish as well as from French and English (and has published translations of books for young readers from Spanish, Catalan, and Italian as well).

Leabhar Breac prints its books in Ireland.

=== Stage and Screen ===

==== Rosg ====
In 1998 along with Ciarán Ó Cofaigh,&lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=http://www.rosg.ie/en/about/History_6/|title=About Us: History|date=July 2025|website=Rosg|access-date=1 August 2025}}&lt;/ref&gt; Ó Scolaí co-founded the film and television production company [http://www.rosg.ie/en/ Rosg] and was co-director until 2006. Rosg produced Ó ScolaÍ’s films ''Cosa Nite'' (1999), ''An Leabhar'' (2001) and ''Na Cloigne'' (2010). He left Rosg in 2006 to devote his time to other artistic activities. 

==== Ealaín ar Oileán ====
In 2004, along with Val Balance, Ó Scolaí co-founded the annual artists' symposium Ealaín ar Oileán (trans., Art on an Island). The Irish-language symposium was held annually in the Áras Éanna arts and cultural center on Inis Oírr ([[wikipedia:Inisheer|Inisheer]], the smallest of the [[wikipedia:Aran_Islands|Aran Islands]]) from 2004 to 2013. Ó Scolaí was its co-director from its founding&lt;ref&gt;{{Cite web|url=https://ga.wikipedia.org/wiki/Darach_Ó_Scolaí.|title=Darach Ó Scolaí|date=3 February 2024|website=Vicipéid|access-date=1 July 2025}}&lt;/ref&gt; until 2013.

Besides being its co-director, Ó Scolaí has taken part in this conference as an artist&lt;ref&gt;{{Cite journal|date=16 January 2005|title=Darach Ó Scolaí|url=https://web.archive.org/web/20050116163252/http://bliainiris.com/authors/darach_oscolai.html|journal=Bliainiris}}&lt;/ref&gt; and writer&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=http://ealainaroilean.ie/ealainaroilean.html|title=The Conference|date=7 September 2013|website=Ealaín ar Oileán|archive-url=https://web.archive.org/web/20130907083744/http://ealainaroilean.ie/ealainaroilean.html|archive-date=7 September 2013|access-date=1 August 2025}}&lt;/ref&gt;.

==== Salamandar ====
In 2006 Ó Scolaí founded the stage production company Salamandar and directed his own play ''An Braon Aníos''. His plays ''An tSeanbhróg'' (2009) and ''Craos'' (2008) were also produced by Salamandar.&lt;ref name=&quot;:19&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/product-category/darach-o-scolai/|title=Darach Ó Scolaí|date=2024|website=Leabhar Breac|access-date=1 July 2025}}&lt;/ref&gt; 

== Works ==

=== Novels ===
* [[wikipedia:An_Cléireach|''An Cléireach'' (trans., ''The Clerk'')]], Leabhar Breac, 2007. The Oireachtas Prize for Literary Fiction, 2007; The Ó Súilleabháin Award (Book of the Year) in 2008, and &quot;named as ‘the best novel since the turn of the Century’ by Comhar.&quot;&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-cleireach/|title=An Cléireach - Leabhar Breac - Irish language novel|website=Leabhar Breac|language=en-US|access-date=2025-10-24}}&lt;/ref&gt;
* ''Na Comharthaí'' (trans., ''The Signs''), Leabhar Breac, 2014.
* ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. The Oireachtas Prize for Literary Fiction, 2019.&lt;ref name=&quot;:4&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/suil-an-daill/|title=Súil an Daill|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Bódléar'', Leabhar Breac, 2024. The Oireachtas Prize for Literary Fiction, 2024&lt;ref name=&quot;:7&quot; /&gt;; The Ó Súilleabháin Award (Book of the Year) in 2025; featured in the 2025 Listen-Up Irish Summer Challenge for students of the Irish language.&lt;ref&gt;{{Cite news|url=https://connachttribune.ie/novel-approach-helps-people-learn-irish-in-a-creative-way/|title=Novel approach helps people learn Irish in a creative way|last=Murphy|first=Judy|date=3 October 2025|work=Connaught Tribune|access-date=24 October 2025}}&lt;/ref&gt;

=== Retellings, Translations and Editions ===
The retellings and translations are into modern Irish.

* ''Feis Tigh Chonáin'' (trans., ''The Feast of Conán's House''), Leabhar Breac, 2000; a retelling of a 15&lt;sup&gt;th&lt;/sup&gt;-century tale from the [[wikipedia:Fenian_Cycle|Fenian Cycle]].&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/feis-tigh-chonain/|title=Feis Tigh Chonáin|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''An Ceithearnach Caolriabhach'' (trans., ''The Narrow-Striped Kern''), Leabhar Breac, 2002; a retelling from c. 1500, also illustrated by Darach Ó ScolaÍ.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-ceithearnach-caolriabhach/|title=An Ceithearnach Caolriabhach|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), Leabhar Breac, 2017, both a modern edition of an 11th-century epic and an annotated edition.&lt;ref name=&quot;:3&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/tain-bo-cuailnge-2-2/|title=Táin Bó Cuailnge|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;  &quot;''Táin Bó Cuailnge'' won the Aodán Mac Poilín Memorial Prize 2017.&quot;&lt;ref name=&quot;:19&quot; /&gt;
* ''Deirdre'', Leabhar Breac, 2023, a “picture book for adults” with artist Anastasia Melnykova.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/deirdre/|title=Deirdre|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; Part of the [[wikipedia:Ulster_Cycle|Ulster Cycle]], ''Deirdre'' is a retelling  of the story of possibly the most widely known Irish figure from the early tales and sagas.&lt;ref&gt;{{Cite book|title=A Dictionary of Celtic Mythology|last=MacKillop|first=James|publisher=Oxford University Press|year=2004|isbn=9780198609674|pages=181}}&lt;/ref&gt;
* ''Bláthnaid'', Leabhar Breac, 2024, a “picture book for adults” with artist Anastasia Melnykova; “one of the great stories of the [[wikipedia:Ulster_Cycle|Ulster Cycle]].”&lt;ref name=&quot;:4&quot; /&gt;
* ''Sadhbh,'' Leabhar Breac, 2025, a picture book for adult readers, illustrated by Alé Mercado; a retelling of the medieval tale ''Ceasacht Inghine Ghuile (''trans., ''The Complaint of Guile's Daughter'').&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/tales-of-wonder/|title=Tales of Wonder|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Eoghan Béal'', Leabhar Breac, 2025, a picture book for adult readers illustrated by Alé Mercado&lt;ref name=&quot;:5&quot; /&gt;; a retelling of the medieval tale ''[https://ga.wikipedia.org/wiki/Caithr%C3%A9im_Cellaig Cathréim Ceallaigh]'' from ''The Yellow Book of Leacan.''&lt;ref name=&quot;:5&quot; /&gt;

=== For Young Readers ===
Ó Scolaí has written illustrated books for young readers (8–10 years old) in two series, the Fionn Series and the Scéalta Staire series, and translated a large number of classics and popular books for children of all ages. The number of these written and translated works suggests a commitment to children and their literacy in Irish. 

The Fionn Series “is a retelling ... of the great legends of the Fianna for the young Irish readers of today.”&lt;ref name=&quot;:6&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/doiteoir-na-samhna/|title=Dóiteoir na Samhna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; [[wikipedia:The_Boyhood_Deeds_of_Fionn|Macgnímartha Finn (The Boyhood Deeds of Fionn)]] is a medieval story in the [[wikipedia:Fenian_Cycle|Fenian Cycle]].

* ''An Bradán Feasa'' (trans., ''The Salmon of Knowledge''), Leabhar Breac, 2010, “shortlisted for the Réics Carlo award 2010.”&lt;ref name=&quot;:9&quot;&gt;{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/an-bradan-feasa/|title=An Bradán Feasa|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Dóiteoir na Samhna'' (trans., ''The Halloween Burner''), 2010.&lt;ref name=&quot;:6&quot; /&gt;
* ''Bodach an Chóta Lachna'' (trans., ''The Churl in the Dun Coat''), 2011.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/|title=Bodach an Chóta Lachna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;

The Scéalta Staire (Historical Stories) series&lt;ref name=&quot;:9&quot; /&gt;

* ''Mánas Ó Dónaill'', 2000.
* ''Seán Ó Néill'', Leabhar Breac, 2000.
* ''Gráinne Mhaol Ní Mháille'', Leabhar Breac, 2003.
* ''Tadhg Dall Ó hUiginn'', Leabhar Breac, 2003.

==== Translations ====

* Robert Louis Stevenson, ''Oileán an Órchiste'' (trans. of ''Treasure Island''), Leabhar Breac, 2014.&lt;ref&gt;{{Cite journal|date=2025-06-19|title=Oireachtas na Gaeilge|url=https://en.wikipedia.org/w/index.php?title=Oireachtas_na_Gaeilge&amp;oldid=1296394643|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Robert Louis Stevenson, ''An Fuadach'' (trans. of ''Kidnapped''), Leabhar Breac, 2016.
* Clement Clarke Moore, ''Cuairt San Nioclás'' (trans. of ''A Visit from St. Nicholas'', or &quot;'Twas the Night Before Christmas&quot;), Leabhar Breac, 2022.
'''''The Corto Maltese Graphic Novels'''''

Written in Italian by Hugo Pratt and translated by Ó Scolaí, both adults and teenagers read this series of Italian adventure graphic novels.&lt;ref&gt;{{Cite journal|date=2025-07-01|title=Corto Maltese|url=https://en.wikipedia.org/w/index.php?title=Corto_Maltese&amp;oldid=1298285365|journal=Wikipedia|language=en}}&lt;/ref&gt; Ó Scolaí's '''translation of ''Corto Maltese''''' was listed in 2017 among &quot;The 30 Irish books that Irish people love.&quot;&lt;ref&gt;{{Cite journal|last=Ó Murchú|first=Eoin P.|date=09/06/2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil [The 30 Irish books that Irish people love]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}&lt;/ref&gt;

* Hugo Pratt, ''Corto: Port na Farraige Goirt'', Leabhar Breac, 2013.
* Hugo Pratt, ''Corto: The Golden House in Samarkand'', 2014.
* Hugo Pratt, ''Corto: Na Liopard-Fhir ó Rufiji'' (trans. of ''Corto: The Leopard Men of Rufiji''), Leabhar Breac, 2014.
* Hugo Pratt, ''Corto: In Ainm Dé Uilthrócairigh'' (trans. of ''Corto: In the Name of God All-Merciful''), Leabhar Breac, 2014.
* Hugo Pratt, ''Corto: Tóraíocht Eile'' (trans. of ''Corto: Another Quest''), Leabhar Breac, 2014.
* Hugo Pratt, ''Corto: Sa tSibéir'' (trans. of ''Corto: In Siberia''), Leabhar Breac, 2016.
'''''Other Translations for Children'''''

Ó Scolaí has translated into Irish six books from the ''Le Pavillon Noir'' (trans., ''Jolly Roger'') series by Alain Surget; four books from the ''Catalan First Steps'' series by Enric Lluch Girbés and the ''Caitlín &amp; Cormac'' series by Joan Carles; three books from the ''Louisette le Taupe'' series by Bruno Heitz, and three books from the ''Loup'' series by Orianne Lallemand.

=== Plays and Screenplays ===

==== Stage Plays ====
Ó Scolaí was writer and director of the original productions of two plays in the ''Trí Bhraon'' (trans., ''Three Drops'') trilogy; ''Coinneáil Orainn'' was directed by Darach Mac Con Iomaire and staged by An Taibhdhearc. All three plays have been published in book form by Leabhar Breac.

* ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32553|title=Coinneáil Orainn|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}&lt;/ref&gt; The first play in the ''Trí Bhraon'' (''Three Drops'') trilogy. [[wikipedia:Taibhdhearc_na_Gaillimhe|An Taibhdhearc]], the national Irish-language theatre of Ireland, toured the country in 2005 with ''Coinneáil Orainn''.&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/coinneail-orainn/|title=Coinneáil Orainn|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; Walter Macken Prize, 2005; BBC Stewart Parker Award, 2006.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/person/darach-scola|title=Darach Ó Scolaí|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}&lt;/ref&gt;
* ''Branwen'', 2006, by Darach Ó Scolaí and Ifor ap Glyn, in Irish, Welsh and English, co-produced by Project Arts Centre and Llwyfan Gogledd Cymru, toured the Republic of Ireland and Wales.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32418|title=Branwen|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt;
* ''An Braon'' Aníos (trans., ''Rising Damp''), 2006, directed by Ó Scolaí.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32461|title=An Braon Aníos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt; The second play in the ''Trí Bhraon'' (''Three Drops'') trilogy. “The Salamandar company toured the country in 2006-07 with this play, and Salamandar also produced a radio version of the play for RTÉ Raidió na Gaeltachta in 2009.”&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/an-braon-anios/|title=An Braon Aníos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''Craos'' (trans., ''Gluttony''), 2008, directed by Ó Scolaí.&lt;ref name=&quot;:13&quot;&gt;{{Cite web|url=https://irishplayography.com/play?playid=32867|title=Craos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt; The third play in the ''Trí Bhraon'' (''Three Drops'') trilogy, it toured to Cork and Belfast.&lt;ref name=&quot;:13&quot; /&gt; A review of the 2008 Salamander performance in the ''Irish Times'' says, “a humorous play which offers plenty to think about, fine acting, and sparklingly witty dialogue.”&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/craos-2/|title=Craos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt;
* ''A+E'', 2008, by Ríonach Ní Néill and Darach Ó Scolaí, &quot;dance and music drama,&quot; co-produced by Ciotóg and Salamandar.&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=32962|title=A+E|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt;
* ''An tSeanbhróg'' (trans., ''The Old Shoe''), 2009, produced by Salamander&lt;ref&gt;{{Cite web|url=https://irishplayography.com/play?playid=33042|title=An tSeanbhróg|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}&lt;/ref&gt; and staged in the Axis Arts Centre, Dublin, and the Letterkenny Arts Centre.
* '''In ''Mhuir Fhíondorcha/The Wine-Dark Sea: The Homer Project'', Ó Scolaí's translation of Homer's Cyclops story, performed at the 2019 IMRAM festival'''.&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/culture/books/imram-a-festival-celebrating-the-irish-language-1.4047610|title=Imram: a festival celebrating the Irish language. Liam Carson reveals the myths and legends appearing in this year’s programme|last=Carson|first=Liam|date=11 October 2019|work=The Irish Times|access-date=15 October 2025}}&lt;/ref&gt;

==== Screenplays ====

* ''Cosa Nite'' (trans., ''Washed Feet''), short film, 1998 (dir. Dearbhla Walsh, prod. Ciarán Ó Cofaigh, Rosg); &quot;a prose version of ''Cosa Nite'' was published (Rosg 2000).&quot;&lt;ref name=&quot;:9&quot; /&gt; Nominated for an Irish Film and Television Award.&lt;ref&gt;{{Citation|title=Cosa Nite (Short 1998) - Awards - IMDb|url=https://www.imdb.com/title/tt0191917/awards/|accessdate=2025-08-25|language=en-US}}&lt;/ref&gt;
* ''Na Glúnta'' (trans., ''The Generations''), 2001&lt;ref&gt;{{Cite web|url=https://www.iftn.ie/production/production_companies/production_sub/feature/?act1=record&amp;aid=70&amp;rid=3917&amp;tpl=filmography_dets&amp;only=1&amp;force=1|title=Na Glúnta {{!}} The Irish Film &amp; Television Network|website=www.iftn.ie|access-date=2025-08-25}}&lt;/ref&gt;, co-directors Ciarán Ó Cofaigh &amp; Darach Ó Scolaí, prod. Ciarán Ó Cofaigh, Rosg.
* ''An Leabhar'' (trans., ''The Book''), short film, 2000, (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg) Rosg, 2000.&lt;ref&gt;{{Citation|title=An Leabhar|url=https://www.imdb.com/title/tt0963767/|publisher=Bord Scannán na hÉireann / The Irish Film Board, ROSG|accessdate=2025-08-25|first=Robert|last=Quinn|others=Colm O&amp;apos;Maonlai, Peadar O&amp;apos;Treasaigh, Diarmuid Mac an Adhastair}}&lt;/ref&gt;
* ''Na Cloigne'' [trans., The Heads], 3-episide series, 2010 (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg), TG4.&lt;ref&gt;{{Cite web|url=https://www.imdb.com/title/tt1607924/|title=Na cloigne|date=2010|website=IMDb|access-date=25 August 2025}}&lt;/ref&gt;
=== Nonfiction ===
Ó Scolaí's essays and lectures are published and his interviews are broadcast regularly, making for a large body of nonfiction critical and analytical work. Here are a few, almost all published in [https://comhar.ie/iris/scribhneoiri/darach-o-scolai/ Comhar]:

* “Ceol Ciúin na nÉagmaise” (trans., “The Silent Music of Absence ['''the Fall?''']”), an essay on the 2014 Nobel Prize winner for literature, [[wikipedia:Patrick_Modiano|Patrick Modiano]], ''Comhar'', December 2014.
* The Ó Cadhain Lecture: [https://leachtaiuichadhain.clo.ie/leachtai/2014 “Cuimhne agus Díchuimhne (trans., “Memory &amp; Forgetfulness&quot;]), 2014.
* “Rithim agus Réim” (&quot;Rhythm and Register&quot;), a public lecture in the University College Dublin lecture series “Ó Thrácht go Twitter” (trans., &quot;From Talk to Twitter&quot;), 2014.
* Review of Pádraig Ó Cíobháin’s ''Dréachta Chrích Fodla'', '''Comhar?, ??'''.
* “Na Geilt i mBun an Tí” (trans., &quot;The Madmen in Charge&quot;), a talk at the Merriman Winter School, Comhar April 2012.&lt;ref name=&quot;:18&quot;&gt;{{Cite web|url=http://darachoscolai.ie/beathaisneis.html|title=Darach Ó Scolaí: Beathaisnéis|website=darachoscolai.ie|access-date=2025-09-26}}&lt;/ref&gt;
* The EFACIS podcast: Síle Ní Choincheannain talks to Darach Ó Scolaí about the historical novel.

== Critical Reception ==
Ó Scolaí’s style has been called “crisp and elegant, and rich in language while being highly readable,”&lt;ref&gt;{{Cite journal|last=Heussaf|first=Anna|date=Summer 2025|title=Bláthnaid—A tale of love, violence and sorcery retold for readers today|url=https://booksirelandmagazine.com/blathnaid-a-tale-of-love-violence-and-sorcery/|journal=Books Ireland}}&lt;/ref&gt; with “an unsurpassed richness and precision of language.”&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Ó Cróinín|first=Breandán|date=Summer 2025|title=unknown|journal=The Limerick Leader}}&lt;/ref&gt; “Whimsical, hilarious, and subtly learned” is how Éilis Ní Dhuibhne described his writing.&lt;ref name=&quot;:20&quot; /&gt;   

=== Original Works ===
Ó Scolaí’s first novel, the 2007 ''An Cléireach'' (''The Clerk'') won two prizes and was described as “one of the great historical novels in the Irish language and among the best books written in the language since the beginning of this century.”&lt;ref name=&quot;:12&quot; /&gt; Novelist Alan Titley says, “In ''An Cléireach'' Ó Scolaí creates the Ireland of war in the 17th century more fully than any other Irish writer on the subject of war since ''L’Attaque'' Eoghain Ó Thuairisc around 1798 [In ''An Cléireach'' cruthaíonn Ó Scolaí Éire an chogaidh san 17ú haois níos iomláine ná mar a dhein aon scríbhneoir Gaeilge eile ar ábhar cogaidh ó ''L’Attaque'' Eoghain Uí Thuairisc timpeall ar 1798].”&lt;ref name=&quot;:11&quot; /&gt;{{rp|25, Col. 1a}} Not all the reviews of this first novel were so positive, however; Proinsias O' Drisceoil says for the Irish Times says,&lt;blockquote&gt;This then is a novel in search of a plot, a story that attempts to attain a significance that eludes it.&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/news/a-disaffected-clerk-in-the-confederates-1.943070|title=A disaffected clerk in the confederates|last=O' Drisceoil|first=Proinsias|date=5 July 2008|work=The Irish Times|access-date=16 October 2025}}&lt;/ref&gt;&lt;/blockquote&gt; 

In the ''Oxford Handbook of Modern Irish Fiction'' Pádraig Ó Siadhail analyzes rather than reviews ''An Cléireach'': &lt;blockquote&gt;In ''An Cléireach'', Ó Scolaí revisits the trauma of Cromwellian Ireland. The primary narrative device is once again the first-hand account, in this case by Tadhg Ó Dúbháin, a clerk and quartermaster in the Confederate Army in 1650. We sample the hardships, the friendships, the tensions, the rivalries, and the petty jealousies amongst comrades in arms, including remnants of the Gaelic literary class, as the Confederate soldiers, increasingly a rabble more than a cohesive unit, retreat in advance of Cromwell’s forces. ''An Cléireach'' concludes with the narrator and his family in exile in continental Europe. But along the retreat route, and central to the novel, members of the Confederate army camp, rest up, and tell versions of a story about the keeper of the treasured manuscript &quot;Saltair an Easpaig&quot; (The Bishop’s Psalter). Their versions raise issues about memory construction, the limitations of individual perspectives, personal agendas, and how minor changes in the telling of a story can alter our understanding of history, Thus, ''An Cléireach'' complements ''Fontenoy'' in moving beyond more realistic recreation of a historical event or period to interrogate the notion of history as construct.&lt;ref&gt;{{Cite book|title=The Oxford Handbook of Modern Irish Fiction|last=Ó Siadhail|first=Pádraig|publisher=Oxford University Press|year=2020|isbn=9780198754893|editor-last=Harte|editor-first=Liam|pages=598–99|chapter=Contemporary Irish Fiction}}&lt;/ref&gt; &lt;/blockquote&gt;

Of ''Súil an Daill,'' in ''Nós'', Cathal Seoighe says, &quot;The book deserves a significant place among the collection of high-quality books published in recent years that would make you feel sorry for someone who does not speak Irish [Tá áit shuntasach ag dul don leabhar i measc an chnuasaigh leabhair ar ardchaighdeán a foilsíodh le roinnt blianta anuas a d’fhágfadh trua agat don té atá gan Ghaeilge].&quot;&lt;ref&gt;{{Cite journal|last=Seoighe|first=Cathal|date=09/26/2022|title=‘Dar leathmhagairle an diabhail, is leabhar den scoth é seo!’ ['According to the devil’s half-wit, this is a great book!’]|url=https://nos.ie/cultur/leabhair/dar-leathmhagairle-an-diabhail-is-leabhar-den-scoth-e-seo/|journal=Nós}}&lt;/ref&gt;

''Bódléar'', Ó Scolaí's most recent book, is a “beautiful novel. There is magic and craftsmanship in it. A small miracle of a book and it is highly recommended.”&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/bodlear-mioruilt-bheag-de-leabhar/|title=Bódléar: Míorúilt bheag de leabhar (Bódléar: A Small Miracle of a Book)|last=Ní Ghairbhí|first=Róisín|date=2024|website=Leabhar Breac|access-date=1 August 2025}}&lt;/ref&gt; Éilis Ní Dhuibhne in the ''Irish Times'' says,&lt;blockquote&gt;what a gem! An affectionately gentle satire of the Irish poetic scene during one creatively fluid 19th-century year, the story focuses on a Maigue poet and schoolteacher who goes on a trip to France and returns with camembert, a cafetiere, ‘Fleurs du Mal’, and a mission to convert the local traditionalists to la modernité. Whimsical, hilarious, and subtly learned, it’s absolutely delightful!&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Ní Dhuibhne|first=Éilis|date=30 June 2025|title=Éilís Ní Dhuibhne on the best Irish language books of 2025 so far:
Including a history of the Gaeltacht Civil Rights Movements, a gem of a novel by Darach Ó Scolaí and Joe McHugh’s entertaining account of learning Irish|url=https://www.irishtimes.com/culture/books/review/2025/06/30/eilis-ni-dhuibhne-on-the-best-irish-language-books-of-2025-so-far/|journal=The Irish Times|pages=22}}&lt;/ref&gt;&lt;/blockquote&gt;

=== Retellings and Translations ===

==== ''Táin Bó Cuailnge'' ====
''Táin Bó Cuailnge'' [''The Cattle Raid of Cooley''] is a modern edition of an 11th-century epic into modern Irish.&lt;ref name=&quot;:3&quot; /&gt; Gearóid Denvir reviewed ''Táin Bó Cuailnge'' for ''Comhar'':&lt;blockquote&gt;Darach Ó Scolaí has ​​achieved a feat in this challenging reworking. He has found a high level of the Irish language to tell his story – as he has done before in his groundbreaking novel An Cléireach (2007, Leabhar Breac) and in his other prose works. This book is a decoration of the language, literature and culture of the Irish language, following the path of the old storytellers and writers and presenting material from the tradition to his own generation according to the understandings of his own time. The book will be a classic that will be of great interest to all readers of the Irish language, both ordinary readers, students, scholars and writers, and there should be a copy in every home in the country. [Tá éacht déanta ag Darach Ó Scolaí san athleagan dúshlánach seo. Tá réim ard den teanga Ghaeilge aimsithe aige lena scéal a inseacht – mar a rinne sé cheana ina úrscéal ceannródaíoch An Cléireach (2007, Leabhar Breac) agus i saothair eile phróis dá chuid. Is maisiú ar an teanga agus ar litríocht agus cultúr na Gaeilge an leabhar seo a leanas conair na seanscéalaithe agus na seanscríobhaithe agus ábhar de chuid an traidisiúin á chur i láthair a ghlúine féin aige de réir thuiscintí a linne féin. Clasaic a bheas sa leabhar a gcuirfidh léitheoirí uilig na Gaeilge, idir ghnáthléitheoirí, mhic léinn, scoláirí agus scríbhneoirí spéis thar na bearta ann, agus ba cheart cóip a bheith i chuile theach sa tír.]&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|last=Denvir|first=Gearóid|date=April 2018|title=Táin Bó Cuailgne|url=https://comhar.ie/iris/78/4/leirmheas/|journal=Comhar|via=JSTOR}}&lt;/ref&gt; &lt;/blockquote&gt;Cathal Poirtéir says, &quot;The freshness and richness of Ó Scolaí’s version are a joy …. The author delights us with the linguistic and stylistic richness of the ancient epic in a modern-Irish version that reflects the original’s spirit and language.&quot;&lt;ref&gt;{{Cite journal|last=Poirtéir|first=Cathal|date=May/June 2018|title=Leabhair Idir Lámha|url=https://www.jstor.org/stable/26564180|journal=Books Ireland|pages=46–47|via=JSTOR}}&lt;/ref&gt;{{rp|47}} Novelist and academic Alan Titley calls Ó Scolaí's &quot;a wonderful gutsy telling&quot; of ''Táin Bó Cuailnge''.&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/culture/2023/03/11/the-tain-retold-maeve-and-ailills-spat-could-be-out-of-a-soap-opera/|title=The Táin retold: ‘Maeve and Ailill’s spat could be out of a soap opera’|last=Titley|first=Alan|date=11 March 2023|work=The Irish Times|access-date=16 October 2025}}&lt;/ref&gt;

==== ''Deirdre'' ====

Marie Whelton, in &quot;Léann Teanga&quot; (&quot;Language Studies&quot;), in the 2024 ''An Reiviú'' says,&lt;blockquote&gt;this version [of ''Deirdre''] by Darach Ó Scolaí succeeds in skillfully capturing and portraying the complexity of gender and power issues in the ‘Deirdre’ tradition [éiríonn leis an leagan seo le Darach Ó Scolaí castacht cheisteanna na hinscne agus na cumhachta i dtraidisiún scéal Dheirdre a ghabháil agus a léiriú go sciliúil]. … There is no doubt that this new version greatly contributes to the legacy of the story and that it revives that legacy thoughtfully and artistically [Níl amhras faoi ach go gcuireann an leagan úr seo go mór le hoidhreacht an scéil agus go ndéanann sé an oidhreacht sin a athbheochan go tuisceanach agus go healaíonta.].&lt;ref name=&quot;:16&quot;&gt;{{Cite web|url=https://www.tara.tcd.ie/tara8/server/api/core/bitstreams/3c20175a-7631-44b2-8b0f-f454edd712b4/content|title=An Artistic Retelling of Deirdre's Tale and the Defeat of Conor Review of Deirdre or the Ship of Mac Uisnigh by Darach Ó Scolaí [Athinsint Ealaíonta ar Oidhe Dheirdre agus ar Ansmacht Chonchúir Léirmheas ar Deirdre nó Loingeas Mhac Uisnigh le Darach Ó Scolaí]|last=Whelton|first=Marie|date=2024|website=The Review [An Reiviú], Language Studies [Léann Teanga]|access-date=25 September 2025}}&lt;/ref&gt;&lt;/blockquote&gt;
=== Works for Young Readers ===
Meadhbh Ní Eadhra said of ''Bodach an Chóta Lachna'' that it was &quot;Beautiful Irish, but easy to understand for young readers.&quot;&lt;ref&gt;Ní Eadhra, Meadhbh. In ''Gaelscéal'', qtd. in &quot;Bodach an Chóta Lachna&quot; https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/.&lt;/ref&gt;

== Awards and Honors ==
Ó Scolaí's works are regularly nominated and make the short list for prizes, an honor in itself, but they are generally not listed here unless they are named as the first-place winner in their category.

=== Oireachtas Prize ===
The Oireachtas Prize is the literary prize awarded by [[wikipedia:Oireachtas_na_Gaeilge|Oireachtas na Gaeilge]], the annual arts festival dedicated to Irish language, arts and culture. Darach Ó Scolaí has won the Oireachtas Prize for Literary Fiction three times, once for ''An Cléireach'' (''The Clerk'') in 2007, for ''Súil an Daill'' (''The Eye of the Blind'') in 2021 and for ''Bódléar'' in 2024.

* 2007, for ''An Cléireach'' (trans., ''The Clerk'') — “(a special prize commemorating the 400th anniversary of the foundation of Coláiste na nGael in Louvain, awarded under the auspices of the Franciscan Province of Ireland). The prize of €10,000 was the largest prize ever awarded to an Irish language novel [(duais speisialta chomórtha 400 bliain bhunú Choláiste na nGael i Lobháin a bronnadh faoi urraíocht Phroibhinse Phroinsiasach na hÉireann). Ba é an duais €10,000 sin an duais ba mhó a bronnadh riamh ar úrscéal Gaeilge].”&lt;ref name=&quot;:18&quot; /&gt;
* 2021, for ''Súil an Daill'' (''The Eye of the Blind'')
* 2024, for ''Bódléar''

=== Ó Shúilleabháin Award, Irish language “Book of the Year” ===
The first prize of this award includes €5,000 to the publisher and €2,500 to the author of the winning work.&lt;ref name=&quot;:10&quot;&gt;{{Cite journal|date=15 August 2023|title=20 saothar san iomaíocht do ‘Leabhair Ghaeilge na Bliana 2023’|url=https://tuairisc.ie/20-saothar-san-iomaiocht-do-leabhair-ghaeilge-na-bliana-2023/|journal=Tuairisc}}&lt;/ref&gt;

* ''An Cléireach'' (''The Clerk'').&lt;ref&gt;{{Cite web|url=http:/www.gaelport.com/uploads/documents/edition19.html|title=Eagrán / Edition 19 - 04 11 2008|date=4/11/2008|website=Internet Archive|archive-url=https://web.archive.org/web/20130525011340/http:/www.gaelport.com/uploads/documents/edition19.html|archive-date=25 May 2013|access-date=25 August 2025}}&lt;/ref&gt;
* ''Táin Bó Cuailnge'', 2018.
* ''Bódléar'', 2025.

==== De Bhaldraithe Award ====
The Gradam de Bhaldraithe is awarded to the best work in translation.&lt;ref name=&quot;:10&quot; /&gt;

* ''Cuairt San Nioclás,'' a translation of Clement Clarke Moore's ''A Visit from St. Nicholas'', or &quot;'Twas the Night Before Christmas.&quot;&lt;ref name=&quot;:10&quot; /&gt;

==== Other ====

* Walter Macken Prize, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005
* Bháiteir Uí Mhaicín Memorial Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005&lt;ref&gt;{{Cite news|url=https://www.irishtimes.com/gaeilge/tuarascail/duais-oireachtais-1.501571|title=Oireachtas Prize: Over €50,000 was awarded to writers in the Oireachtas Literary Competitions at an event in Dublin last night. Winners… [Duais Oireachtais: Bronnadh breis agus €50,000 ar scríbhneoirí i gComórtais Liteartha an Oireachtais ar ócáid i mBaile Átha Cliath aréir. Bhuaigh…]|work=5 October 2005|access-date=15 October 2025}}&lt;/ref&gt;
* BBC Stewart Parker Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2006
* The Aodán Mac Póilín Commemorative Prize, for ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), 2017

== External Links ==

* Leabhar Breac website: https://leabharbreac.com/en/
* Leabhar Breac Facebook pages:
* Rosg website: [http://www.rosg.ie/en/ &lt;nowiki&gt;http://ww&lt;/nowiki&gt;w.rosg.ie/en/]
* Art on the Island (Ealaín ar Oileán) website, archived at the Wayback Machine: https://web.archive.org/web/20130601000520/http://ealainaroilean.ie/ 31 March 2012, 1 June 2013 and 8 January 2014

* Darach Ó Scolaí's website Archived 25 September 2015 at the Wayback Machine: https://web.archive.org/web/20150925103456/http://darachoscolai.ie/
* Youtube video of [https://www.youtube.com/watch?v=OlP2AmSBzXc Breandán Ó Cróinin introducing Deirdre at the book launch] in the pub Tigh Mholly (Molly’s House).

== Primordial Ooze ==

* Known for his sensitivity to language and voices.
* Finish scanning through JSTOR
* Scan through Irish Times, 56 hits
* Check Goodreads
* Check YouTube (In the spring of 2013, the arts programme Imeall interviewed the author on TG4.)
* Check both Wikipedias for pages on the origins of the retold tales (like Deirdre) and link to this article
* Propose link from University of Galway page once Darach’s is up
* Write Irish National Biography (&lt;nowiki&gt;https://www.dib.ie&lt;/nowiki&gt;) to propose an article about Darach once the Wikip article is done? See what they say.
* Link to Ó Scolaí from the Wikipedia
* Make sure links '''to''' Wikipedia in the actual encyclopedia work right

=== Not Placed Yet ===

* &quot;So here are the books that Irish people love the most! [Mar sin seo iad na leabhair is gile leis na Gaeil!]&quot; — &quot;32. An Cléireach – Darach Ó Scolaí (2)&quot; [18 books got 2 votes, and then they're alphabetized by author's last name, so the 32 of 34 doesn't signify the specificity it seems to]&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Ó Murchú|first=Eoin P.|date=9 June 2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil. [The 30 best Irish books for Irish people]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}&lt;/ref&gt;
* &quot;Below is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers.Here is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers [Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí.Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí].&quot; &quot;Corto Maltese – Hugo Pratt (aistrithe ag Darach Ó Scolaí)&quot;&lt;ref name=&quot;:14&quot; /&gt;
* &quot;Ceann eile de bhuaicphointí na hÉigse a bheidh sa seisiún le Darach Ó Scolaí, duine d’úrscéalaithe móra na Gaeilge, agus duine de chomhbhunaitheoirí teach foilsitheoireachta Leabhar Breac. [Another highlight of the Éigse will be the session with Darach Ó Scolaí, one of the great Irish language novelists, and one of the co-founders of the publishing house Leabhar Breac.]&quot;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Nós|date=4 May 2023|title=Éigse na Bruiséile le filleadh i mí na Bealtaine. [Éigse na Bruséile to return in May]|url=https://nos.ie/cultur/eigse-na-bruiseile-le-filleadh-i-mi-na-bealtaine/|journal=Nós}}&lt;/ref&gt;

=== Things Taken Out for Now ===
“’The play is a comedy about language, lies, bureaucracy and Gaeltacht grants, in the tradition of Myles na Gcopaleen,’ according to Norma-Jean Kenny in the ''Galway Advertizer'', ‘in which the author comments and criticizes the institutions of the Irish language in Ireland without ceasing.’&quot;

Supposedly a quotation by Gearóid Denvir reviewing ''Táin Bó Cuailnge'' for ''Comhar'' (but I don't find it in the article):

This book has long been needed by Irish language readers and there is no doubt that it will become a classic in time and surpass Thomas Kinsella’s English version. This version remains faithful to the language of the original while at the same time finding an appropriate language in today’s Irish. Ó Scolaí masterfully overcomes the difficulties of the original’s rhetorical difficulties and the versions of the original poetic texts are extremely effective.[supposedly &lt;ref name=&quot;:15&quot; /&gt;]

“The biggest prize ever awarded for a novel in Irish was presented at a special ceremony in the National Concert Hall in Dublin, today (Thursday, 4 October 2007). Darach Ó Scolaí, writer, artist &amp; playwright from Casla, Co. Galway, was awarded €10,000 for his literary novel, ‘An Ardscoil’. This work, under the new title ‘An Cléireach’, will be launched at Oireachtas na Samhna in Westport in November. This is the first novel from his pen, a story set in the late seventeenth century. This competition was sponsored by the Franciscan Province of Ireland.” (archive, Oireachtas na Gaeilge site, 04 October, 2007)

''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. number 2 in ''Comhar'' literary magazine’s list of best books of 2021. '''{6}.'''

*William Shakespeare, ''Romeo agus Juliet'' (trans. of ''Romeo and Juliet''), Leabhar Breac, 2016.
*Jonathan Swift, ''Camchuairt Ghuilivéir'' (trans. of ''Gulliver's Travels''), Leabhar Breac, 2016.

*Hugo Pratt, ''Corto Maltese''


'''''Flag of Bones (Bratach na gCnámh) Series'''''

Leabhar Breac published the Bratach na gCnámh series of books for young readers. Written in French by Alain Surget, illustrated by Annette Marnat and translated by Darach Ó Scolaí, this series uses the history of Caribbean Sea pirates&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/product-category/alain-surget/|title=Alain Surget Archives|website=Leabhar Breac|language=en-US|access-date=2025-09-30}}&lt;/ref&gt;:

*Alain Surget, ''Éalú as Páras'' (''Escape from Paris''), Annette Marnat (Illustr.), Leabhar Breac, 2011.
* Alain Surget, ''Oilean na Siorcanna'' (''Shark Island''), Annette Marnat (Illustr.), Leabhar Breac, 2011.
* Alain Surget, ''Long na dTaibhsi'' (''Ship of the Ghosts''), Annette Marnat (Illustr.), Leabhar Breac, 2011.
* Alain Surget, ''San Ochtapas Dubh'' (''In the Black Octopus''), Annette Marnat (Illustr.), Leabhar Breac, 2013.
* Alain Surget, ''San Ionsai ar Veracruz'' (''The Attack on Veracruz''), Annette Marnat (Illustr.), Leabhar Breac, 2013.
'''''For &quot;First Readers&quot; (children to 6 years old or so)'''''

These books were written originally in Catalan by Spanish author Enric Lluch Girbés and translated into Irish by Ó ScolaÍ:
*Enric Lluch, ''Ag Péinteáil an Tí'' (''Painting the House''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''An Colúr Bacach'' (''The Lazy Dove''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''An Phluais'' (''The Cave''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''Madra Dhaideo'' (''Grandpa's Dog''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
*Enric Lluch, ''Fiacail Mháire'' (''Mary's Tooth''), Anna Clariana (Illustr.), Leabhar Breac, 2017.
'''''Bruno Heitz'''''

Leabhar Breac published a series of 3 Heitz books for small children. Published originally in French, this series of three comic books is about a blind mole named Cáitín Chaoch in Irish (and ''Louisette la taupe'' in French).&lt;ref&gt;{{Cite web|url=https://leabharbreac.com/en/product-category/bruno-heitz-en/|title=Bruno Heitz Archives|website=Leabhar Breac|language=en-US|access-date=2025-10-02}}&lt;/ref&gt; Ó Scolaí translated these:

*Bruno Heitz (author and illustr.), ''Práinneach'' (''Urgent''), Leabhar Breac, 2020
*Bruno Heitz (author and illustr.), ''Preab san Aer'' (''Bounce in the Air''), Leabhar Breac, 2020.

'''''Books for Toddlers'''''

Leabhar Breac has published 14 books written by French author Orianne Lallemand's and illustrated by Eleonore Thuillier, about Lallemmand's popular character Loup, Wolf. These are translated by Ó Scolaí:
*Orianne Lallemand, ''An Mac Tire a Raibh Faitios an Domhain Air'' (trans. of ''The Son Who Saw the World in His Eyes''), Eleonore Thuillier  Illustr.), Leabhar Breac, 2018.
*Orianne Lallemand, ''Macan agus an Goban'' (trans. of ''Macan and the Goblin''), Eleonore Thuillier (Illustr.), Leabhar Breac, 2018.
* Orianne Lallemand, ''A Mac Tíre a Chuaigh go Tóin na Farraige'' (trans. of ''The Wolf Who Went to the Bottom of the Sea''), Éléanore Thuillier  (Illustr.), Leabhar Breac, 2019.
'''''Board Books (for babies)'''''

J. C. (Joan Carles) Girbés Aparisi is a Catalan author and editor. These books were written in Catalan and translated by Ó Scolai.
*J. C. Girbés, ''An Phicnic'' (''The Picnic''), Silvia Ortega (Illustr.), Leabhar Breac, 2013.
* J. C. Girbés, ''An Chóisir'' (''The Party''), Silvia Ortega (Illustr.), Leabhar Breac, 2013.
*J. C. Girbés, ''Lá Mór Fada'' (''A Long Day''), Silvia Ortega (Illustr.), Leabhar Breac, 2014.
*J. C. Girbés, ''Tabhair Leat do Leabhar'' (''Bring Your Book''), Silvia Ortega (Illustr.), Leabhar Breac, 2014.

==== Gradam Réics Carló ====
The Réics Carló prize is awarded for the best book in the Irish language for young readers. It is named for one of the characters of 20th-century writer [[wikipedia:Cathal_Ó_Sándair|Cathal Ó Sándair (Charles Saunders)]].

* ''An Bradán Feasa'' was “shortlisted for the Réics Carlo award 2010.”&lt;ref name=&quot;:9&quot; /&gt;

== References ==
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    <title>Universal Bibliography/Countries</title>
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See also [[Universal Bibliography/Geography|Geography]].

See [[w:Category:Bibliographies of countries or regions]] and [[w:Category:Works about countries]].

This part of the [[Universal Bibliography]] is a bibliography of countries (including former countries).

==Countries==
*Bateman and Egan (eds). The Encyclopedia of World Geography: A Country by Country Guide. 1993. Revised 1997.
*Peter Stalker. Handbook of the World. 2000. A Guide to Countries of the World. (Oxford Guide to Countries of the World. 2nd Ed: 2004, 2nd Revised Ed: 2007 [https://books.google.co.uk/books?id=GtztAAAAMAAJ], 3rd Ed: 2010 [https://books.google.co.uk/books?id=gvKvfxkbZ1AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Countries of the World and Their Leaders Yearbook. Gale. [https://books.google.co.uk/books?id=5etKAAAAYAAJ] [https://books.google.co.uk/books?id=p41OAAAAIAAJ]
*Hutchinson Guide to Countries of the World [https://books.google.co.uk/books?id=GgpjUe4kN_IC]
*The World Guide: Global Reference, Country by Country. 11th Ed: 2007 [https://books.google.co.uk/books?id=EoWoLgAACAAJ]
*Spence. The World Today: A Nation-by-Nation Guide. Cassell. 1994. 1999. [https://books.google.com/books?id=Ub8qOQAACAAJ]
*Worldmark Encyclopedia of the Nations [https://books.google.co.uk/books?id=I0oYAQAAMAAJ]
*Kurian. Encyclopedia of the World's Nations. Facts on File. Reviews: [https://books.google.co.uk/books?id=Y1EnAQAAIAAJ] [https://books.google.co.uk/books?id=lz0RAQAAMAAJ]
*Michael O'Mara. Facts about the World's Nations. 1999. [https://books.google.co.uk/books?id=mygYAAAAIAAJ]
*Status of the World's Nations. 1965 [https://books.google.co.uk/books?id=sftEyRbAXMUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 1973 [https://books.google.co.uk/books?id=kw2U_Cg2gKYC&amp;pg=PP3#v=onepage&amp;q&amp;f=false].
*[[s:Author:John Alexander Hammerton|Hammerton, John Alexander]] (ed). Countries of the World. Published at the Fleetway House. 6 vols. [https://books.google.co.uk/books?id=e6IaAQAAMAAJ] [https://books.google.co.uk/books?id=K5oaAQAAMAAJ]
*[[s:Author:Robert Brown (1842-1895)|Brown, Robert]]. The Countries of the World. [https://books.google.co.uk/books?id=nO0DAAAAQAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*A Morely Dell. The Countries of the World. (Harrap's New Geographical Series). 1932. (School certificate). Reviews: [https://books.google.co.uk/books?id=oSS9PB_Jf7AC] [https://books.google.co.uk/books?id=BicVAAAAIAAJ] [https://books.google.co.uk/books?id=5qBOAAAAIAAJ] [https://books.google.co.uk/books?id=YbwcAQAAIAAJ] [https://books.google.co.uk/books?id=sc1AAAAAIAAJ]

General series:
*National Geographic Countries of the World [https://books.google.co.uk/books?id=IT2wfzVIPykC]
*Countries of the World. Evans Brothers. (GCSE) [https://books.google.co.uk/books?id=a3sZvWc7E1EC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*One Europe. Longman. [https://search.worldcat.org/en/title/west-germany-adapted-by-lj-russon-from-the-original-german-by-sylvia-lof-ingrid-mallberg-dietrich-rosenthal/oclc/561591761]
*Collier's Nations of the World. The Nations of the World: An Historical Series. [https://books.google.co.uk/books?id=VJY-AAAAYAAJ&amp;pg=PP8#v=onepage&amp;q&amp;f=false]
*Collier's History of Nations. The History of Nations. [https://books.google.co.uk/books?id=fmSUfTY5E80C]
*The Story of the Nations. T Fisher Unwin.
*The World and Its Peoples. (The Illustrated Library of the World and Its Peoples). Greystone Press, New York.
*World and Its Peoples. Marshall Cavendish. [https://books.google.co.uk/books?id=oms5xjI7ba0C&amp;pg=PA141#v=onepage&amp;q&amp;f=false]

==England==
===Counties===
See [[s:Portal:Counties]]

* Harrison, &quot;County Bibliography&quot; (1886) 3 Library Chronicle [https://books.google.co.uk/books?id=Wz9FAAAAYAAJ&amp;pg=PA49#v=onepage&amp;q&amp;f=false 49]

General series
*Victoria County History. [Review: [https://archive.org/details/sim_athenaeum-uk_1913-01-04_4445/page/5/mode/1up].]
*Oxford County Histories
*Pinnock's County Histories
*Shire County Guides. Shire Publications.
*Cambridge County Geographies
*Pike's New Century Series
*[[s:Page:County Churches of Cornwall.djvu/6|County Churches]]. G Allen.

Avon
*Moore. Avon Local History Handbook. Phillimore. 1979. [https://books.google.co.uk/books?id=h0kjAAAAMAAJ] Bibliography, p 102

Bedfordshire
*Conisbee, Lewis Ralph. A Bedfordshire Bibliography. Bedfordshire Historical Record Society. Bedford. 1962. Supplements 1967, 1971, 1978. Third supplement by Threadgill. Review: 6 Archives 52 [https://books.google.co.uk/books?id=oOMZAAAAYAAJ]. See also [https://books.google.co.uk/books?id=MjspAAAAYAAJ] [https://books.google.co.uk/books?id=PejgAAAAMAAJ]
*Godber. History of Bedfordshire. 1969. 1984. [https://books.google.co.uk/books?id=jdvwPQAACAAJ]
*Pinnock. The History and Topography of Bedfordshire [https://books.google.co.uk/books?id=9bJYAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Parry. Select Illustrations, Historical and Topographical, of Bedfordshire [https://books.google.co.uk/books?id=UTUJAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Blyth. The History of Bedford and Visitor's Guide. 1873 [https://books.google.co.uk/books?id=IuIGAAAAQAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Cambridge County Geographies [https://books.google.co.uk/books?id=kTc8AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Buckinghamshire
*Reed. A History of Buckinghamshire. 1993 [https://books.google.co.uk/books?id=BtkWAQAAIAAJ]

Cambridgeshire
*Carter. History of the County of Cambridge [https://books.google.co.uk/books?id=jXpbAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Babington. Ancient Cambridgeshire [https://books.google.co.uk/books?id=DPrCAwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Devon
*Ravenhill and Rowe. Devon Maps and Map-makers [https://books.google.co.uk/books?id=tjf2yAEACAAJ]
*Wright. A Plea for a Devonshire Bibliography. 1885 [https://books.google.co.uk/books?id=8ZUDAAAAQAAJ]

Derbyshire
*Woore. A Catalogue of Local Maps of Derbyshire, C.1528-1800. 2012. [https://books.google.co.uk/books?id=oWmCMwEACAAJ]
*O'Neal. A Bibliography of Derbyshire Lead Mining. 1961

Essex
*Cunnington. Catalogue of Books, Maps and Manuscripts, relating to or connected with the County of Essex. 1902 [https://books.google.co.uk/books?id=oIcqpibGE4MC]
*&quot;The Bibliography of Essex&quot; (1882) 1 Antiquarian Magazine &amp; Bibliographer [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false 72]. See also [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA283#v=onepage&amp;q&amp;f=false p 283].
*&quot;The Bibliography of Essex&quot; (1891) 5 The Essex Naturalist 30 [https://books.google.co.uk/books?id=iIo1AQAAMAAJ]
*Moon. Essex Literature. 1900. Review: 61 Literary World 438 [https://books.google.co.uk/books?id=2T0ZAAAAYAAJ] See also [https://books.google.co.uk/books?id=1Y4UAQAAIAAJ] [https://books.google.co.uk/books?id=C_pEAAAAMAAJ]
*Fenn and Lowery, &quot;An Essex Bibliography&quot;, Journal of the South West Essex Technical College, vols 2 &amp; 3
*Victoria County History bibliography. 1959 [https://books.google.co.uk/books?id=F2EJAQAAIAAJ]
*O'Leary, John Gerard. A Supplement to the Essex Bibliography. Dagenham. 1962.
*A Bibliography of Essex Archaeology &amp; History
*Essex and Dagenham: A Catalogue of Books, Pamphlets and Maps. Dagenham. 1961
*Essex Archaeology and History: The Transactions of the Essex Society for Archaeological and History [https://books.google.co.uk/books?id=CtFAAAAAYAAJ]
*Essex Naturalist: Being the Journal of the Essex Field Club
*Wright. The History and Topography of the County of Essex [https://books.google.co.uk/books?id=SgQVAAAAQAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Ogborne, The History of Essex [https://books.google.co.uk/books?id=IeVSAAAAcAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Suckling. Memorials of the Antiquities and Architecture, Family History and Heraldry of the County of Essex [https://books.google.co.uk/books?id=bcw_AAAAcAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Hunter, The Essex Landscape: A Study of Its Form and History [https://books.google.co.uk/books?id=w9kWAQAAIAAJ]
*Cambridge County Geography [https://books.google.co.uk/books?id=GPHa_X_0qo0C&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Sokoll. Essex  Pauper Letters, 1731-1837 [https://books.google.co.uk/books?id=rCLia7XlqtMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Morant. The History and Antiquities of Colchester in the County of Essex [https://books.google.co.uk/books?id=DDgtAAAAYAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Wallen. The History and Antiquities of the Round Church at Little Maplestead, Essex [https://books.google.co.uk/books?id=FPYVAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Kent
*Smith. Bibliotheca Cantiana. 1837. [https://books.google.co.uk/books?id=1dJDAAAAYAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false]

Leicestershire
*Kirkby, C V (compiler). Catalogue of the books, pamphlets, &amp;c., relating to Leicestershire in the Central Reference Library. Leicester Free Public Libraries. 1893. Reviews: [https://books.google.co.uk/books?id=3boqAQAAIAAJ&amp;pg=PA84#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=UcHnAAAAMAAJ&amp;pg=PA728#v=onepage&amp;q&amp;f=false]
*Leicestershire and Rutland Bibliography, 1963-65 (1966) [https://books.google.co.uk/books?id=-OhVAAAAYAAJ 40] Leicestershire Archaeological and Historical Society: Transactions (1964/5) 92. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1961-63. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1960-61. Available as pdf from University of Leicester.
*A Bibliography of the Small Towns in Leicestershire and Rutland, 1600–1850. (Dissertation). [https://repository.lboro.ac.uk/articles/educational_resource/A_bibliography_of_the_small_towns_in_Leicestershire_and_Rutland_1600_1850/9414200]
*Loughborough's Heritage: A Bibliography of the Holdings of Leicestershire Libraries and Information Service and Record Office. [https://books.google.co.uk/books?id=Bwx2zgEACAAJ]
*Keith Ambrose and Frank Williams, &quot;Bibliography of the Geology of Leicestershire and Rutland: Part 2: 1971-2003&quot; (2004) [https://books.google.co.uk/books?id=U-tQAQAAIAAJ 16] The Mercian Geologist 5. Available as pdf from East Midlands Geological Society.
*Parsons and Brandwood. A Bibliography of Leicestershire Churches. 1978.
*Education in Leicestershire: A Bibliography. [https://books.google.co.uk/books?id=X6EfzQEACAAJ]

Sussex
*Brent, Fletcher and McCann. Sussex in the 16th and 17th Centuries: A Bibliography. 2nd Ed [https://books.google.co.uk/books?id=I7UtAAAAYAAJ]
*Farrant. Sussex in the 18th and 19th Centuries: A Bibliography. 1st Ed: 1973, 2nd Ed: 1977 [https://books.google.co.uk/books?id=MLUtAAAAYAAJ], 3rd Ed: 1979

==France==
Bibliography:
*Bibliographie de la France. Commentary: Encyclopedia of Library and Information Science, vol 37, supplement 2, [https://books.google.co.uk/books?id=10rgjNvOV8oC&amp;pg=PA145#v=onepage&amp;q&amp;f=false p 145]; The Bookseller, 6 January 1881, [https://books.google.co.uk/books?id=4dsiAQAAMAAJ&amp;pg=PA10#v=onepage&amp;q&amp;f=false p 10]; Stein, Manuel de bibliographie générale, [https://books.google.co.uk/books?id=lJYPyKjV1qYC&amp;pg=PA23#v=onepage&amp;q&amp;f=false p 23]. 
*Girault de Saint-Fargeau. Bibliographie historique et topographique de la France. 1845 [https://books.google.co.uk/books?id=kClB9CQNZoMC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Catalogue d'une collection d'ouvrages sur l'histoire des provinces de la France. 1842 [https://books.google.co.uk/books?id=qQBX5WZouzAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Landscape:
*Beaujeu-Garnier. France. (The World's Landscapes). 1975. [https://books.google.com/books?id=nwxDAQAAIAAJ]

Agenais:
*Andrieu. Bibliographie générale de l’Agenais et des parties du Condomois et du Bazadais. 1886 to 1891. Reprinted 1969.

Alsace:
*Ristelhuber. Bibliographie alsacienne. 1869 to 1873 [https://books.google.co.uk/books?id=0mhLAQAAMAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Bibliographie alsacienne: Revue critique des publications concernant l'Alsace. 1918 to 1936
*Ritter. Répertoire bibliographique des livres imprimés en Alsace aux XVe et XVIe siècles [https://books.google.co.uk/books?id=DewaAQAAMAAJ]

Angoumois:
*Castaigne. Essai d'une bibliothèque historique de l'Angoumois, ou Catalogue raisonné des principaux ouvrages qui traitent des différentes branches de l'histoire de cette province. 1847 [https://books.google.co.uk/books?id=R-UanmmlvAEC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Anjou:
*Braguier and Braguier. Archéologie en Anjou: bibliographie. 1984 [https://books.google.co.uk/books?id=LvsmAQAAIAAJ]

Auvergne:
*Gonot. Catalogue des ouvrages imprimés et manuscrits concernant l'Auvergne, extrait du catalogue général de la Bibliotlèque de Clermont-Fd (Puy-de-Dome). 1849. [https://books.google.co.uk/books?id=yCFtbObRCbUC&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Catalogue des livres et estampes concernant l'ancienne Province d'Auvergne (Puy-de-Dôme, Cantal, Haute-Loire) réunis par feu M. G. Desbouis. 1865. [https://books.google.co.uk/books?id=Ui4S8_D0N74C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Béarn
*&quot;Bibliographie Béarnaise&quot;, Revue de Pau et du Béarn [https://books.google.co.uk/books?id=FuZnAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=FQYqvPo9D9IC&amp;pg=PA158#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RL9VAAAAYAAJ]

Brittany
*Sacher. Bibliographie de la Bretagne, ou Catalogue général des ouvrages historiques, littéraires et scientifiques parus sur la Bretagne, avec la liste des revues publiées en cette province, les prix approximatifs des volumes rares, etc. 1881 [https://archive.org/details/bibliographiede00sach]

Burgundy:
*Milsand. Bibliographie bourguignonne; ou, Catalogue méthodique d'ouvrages relatifs à la Bourgogne: Sciences - Arts - Histoire. 1885 [https://archive.org/details/bibliographiebo00milsgoog] [https://archive.org/details/bibliographiebo00sciegoog] [https://books.google.co.uk/books?id=CxIIAAAAQAAJ]
*Catalogue des manuscrits de la Bibliothèque royale des ducs de Bourgogne. 1842 [https://books.google.co.uk/books?id=FX5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*The Companion Guide to Burgundy [https://books.google.co.uk/books?id=NraRP0AkDT0C&amp;pg=PP3#v=onepage&amp;q&amp;f=false]
*Lecat. The Golden Book of Burgundy. (The Golden Book) [https://books.google.co.uk/books?id=FyzR9qU1Zl4C&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Gwynn. Burgundy: With Chapters on the Jura and Savoy. (Kitbag Travel Books). 1935 [https://books.google.co.uk/books?id=ny1LAAAAMAAJ]
*Bazin. Wonderful Burgundy. 1988. 1997 [https://books.google.co.uk/books?id=Yt1CRdICWCUC]
*Bailey. Burgundy. (Insight Guides). 1993 [https://books.google.co.uk/books?id=Q69a1dMW2NQC]
*Dunlop. Burgundy. Hamilton.1990 [https://books.google.co.uk/books?id=S_1OAAAAMAAJ]

Champagne:
*Lhermitte. Ouvrages sur la Champagne: contribution à la bibliographie champenoise. 1992. [https://books.google.co.uk/books?id=jbPfAAAAMAAJ]

Dauphiné:
*Mélanges biographiques et bibliographiques relatifs à l'histoire littéraire du Dauphiné par Colomb de Batines et Ollivier Jules. 1837 [https://books.google.co.uk/books?id=2F5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Lorraine:
*Bibliographie lorraine. Académie nationale de Metz [https://books.google.co.uk/books?id=n-DfAAAAMAAJ]

Maine:
*Desportes. Bibliographie du Maine, précédée de la description topographique et hydrographique du diocése du Mans, Sarthe et Mayenne. 1844. [https://books.google.co.uk/books?id=hSk-AAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Normandy:
*Frère. Manuel du bibliographe Normand ou dictionnaire bibliographique et historique. 1858 to 1860. [https://books.google.co.uk/books?id=dp6geJClg1YC&amp;pg=PP13#v=onepage&amp;q&amp;f=false vol 1]

==Japan==
Bibliography and literature
*Hideo Kaneko. &quot;Japanese Literature and Bibliography&quot;. Kent, Lancour and Daily (eds). Encyclopedia of Library and Information Science. Marcel Dekker. 1977. vol 21. pp [https://books.google.co.uk/books?id=H1pNvzr_n98C&amp;pg=PA131#v=onepage&amp;q&amp;f=false 131] to 176.

Bibliography
*Jozef Rogala. A Collector's Guide to Books on Japan in English: An Annotated List of Over 2500 Titles with Subject Index. 2001. [https://books.google.co.uk/books?id=7KI9ao-w2FEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ria Koopmans-de Bruijn. Area Bibliography of Japan. (Scarecrow Area Bibliographies). Scarecrow Press. 1998. [https://books.google.co.uk/books?id=Hlx2OMjgUi0C&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Frank Joseph Shulman. Japan. (World Bibliographical Series, vol 103). Clio Press. 1989. [https://books.google.co.uk/books?id=LsoUAQAAIAAJ]
*Eibun Nihon Kankei Tosho Mokuroku, 1945-1981. (Japanese: 英文日本関係図書目録, 1945-1981). (English: Catalogue of Books in English on Japan, 1945-1981). Japan Foundation. Tokyo. 1986.
*Japan: analytical bibliography: with supplementary research aids: and selected data on Okinawa . . . Department of the Army. Washington. 1972. [https://books.google.co.uk/books?id=h4d4nYxrxtMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Books on Japan in Western Languages. The International Christian University Library. 1971. [https://books.google.co.uk/books?id=F2bQAAAAMAAJ]
*Books on Japan: A List of Acquisitions, 1955-1970. International House of Japan Library. 1971. [https://books.google.co.uk/books?id=F8sWAQAAIAAJ]
*Fukuda. Union Catalog of Books on Japan in Western Languages. 1968. [https://books.google.co.uk/books?id=HKYyAQAAIAAJ]
*A Classified List of Books in Western Languages Relating to Japan. University of Tokyo Press. 1965. [https://books.google.co.uk/books?id=U8MUAQAAIAAJ]
*Katsuji Yabuki (ed). Japan Bibliographic Annual. Published by the Hokuseido Press for the Japan Writers Society. 1956 and 1957.
**Japan Bibliographic Annual 1956. [https://books.google.co.uk/books?id=9XLQAAAAMAAJ]
**Japan Bibliographic Annual 1957. [https://books.google.co.uk/books?id=vesSAAAAIAAJ]. Reviews: (1957) 13 Monumenta Nipponica 166 (April-July) [https://books.google.co.uk/books?id=8S1yb-iwrOwC] (1957) 25 The Oriental Economist 212 (April) [https://books.google.co.uk/books?id=QELoAAAAMAAJ]
*Haring. Books on Japan: A Reference List. 1955. [https://books.google.co.uk/books?id=RbDoAAAAMAAJ]
*Borton. A Selected List of Books and Articles on Japan in English, French, and German. 1940: [https://books.google.co.uk/books?id=YYIsAAAAYAAJ]. Revised and enlarged. Harvard University Press. 1954: [https://books.google.co.uk/books?id=F8O2VwJUPUkC].
**A Selected List of Books on Japan in Western Languages (1945-1960). (Studies on Asia Abroad, vol 1). The Information Centre of Asian Studies, The Toyo Bunko. 1964. [https://books.google.co.uk/books?id=i1_QAAAAMAAJ]
*Oskar Nachod. Bibliography of the Japanese Empire 1906-1926. 1928. [https://archive.org/details/bibliographyofja0001oska/page/n8/mode/1up vol 1]. [https://archive.org/details/bibliographyofja0002oska/page/n6/mode/1up vol 2].
*Fr. von Wenckstern. A Bibliography of the Japanese Empire: being a Classified List of All Books, Essays and Maps in European Languages relating to Dai Nihon (Great Japan) published in Europe, America and in the East from 1859-93 . . . 1895. vol 1. [https://books.google.co.uk/books?id=dcVAAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=v7lO4ddqDywC&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**Volume 2, from 1894 to the middle of 1906. 1907. [https://archive.org/details/bibliographyofja0002frvo/page/n6/mode/1up]
*Hyman Kublin. What Shall I Read on Japan? An Introductory Guide. Japan Society, New York. 1971. [https://books.google.co.uk/books?id=yRRUAAAAYAAJ]
*Henri Cordier. Bibliotheca Japonica: Dictionnaire Bibliographique des Ouvrages Relatifs a L'Empire Japonais. 1912. [https://archive.org/details/bibliothecajapon0008henr/page/n8/mode/1up]

Japanese studies
*An Introductory Bibliography for Japanese Studies. The Japan Foundation. [https://books.google.co.uk/books?id=53O6AAAAIAAJ]
*Richard Perren. Japanese Studies from Pre-History to 1990: A Bibliographical Guide. 1992. [https://books.google.co.uk/books?id=CN9RAQAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies&quot; at pp 1 to 3.
*K.B.S. Bibliography of Standard Reference Books for Japanese Studies, with Descriptive Notes. University of Tokyo Press. [https://books.google.co.uk/books?id=95wbAAAAMAAJ]
*[[w:en:Monumenta Nipponica|Monumenta Nipponica]]
*[[w:en:Japan Forum|Japan Forum]]. British Association for Japanese Studies. [https://www.tandfonline.com/journals/rjfo20]
*Japanese Studies. Japanese Studies Association of Australia. ISSN 1037-1397. [https://www.tandfonline.com/journals/cjst20]
*[[w:en:The Journal of Japanese Studies|The Journal of Japanese Studies]]. University of California Press.
*Nichibunken Newsletter. [[w:en:International Research Center for Japanese Studies|International Research Center for Japanese Studies]]. [https://books.google.co.uk/books?id=5TnjAAAAMAAJ] [https://www.nichibun.ac.jp/en/publications/data/news/] [https://newsletter.nichibun.ac.jp/en/]

History and culture
*John W Dower. Japanese History &amp; Culture from Ancient to Modern Times: Seven Basic Bibliographies. 1986. [https://books.google.co.uk/books?id=NX67AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies &amp; Research Guides&quot; at chapter 6.

Research guides
*Mindy L Kotler. Information Gathering on Japan: A Primer. Search Associates. 1988. ISBN 9780962546006. Catalogue: [https://search.worldcat.org/zh-cn/title/Information-gathering-on-Japan-Joho-:-a-primer/oclc/20530148]. Review: (1989) [https://books.google.co.uk/books?id=NZLiAAAAMAAJ 27] Choice 82

Encyclopedias

See also [[w:ja:Japanese encyclopedias]]
*Louis Frédéric. [[w:Japan Encyclopedia|Japan Encyclopedia]]. Translated by Käthe Roth. 2002. [https://books.google.co.uk/books?id=p2QnPijAEmEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
**Louis Frédéric. Le Japon: Dictionnaire et Civilisation. Robert Lafont. Paris. 1996. ISBN 2-221-06764-9.
*Japan: An Illustrated Encyclopedia. Kodansha. 1993.
**Japan: Profile of a Nation. Kodansha. 1995. Revised Edition. 1999.
*[[w:Kodansha Encyclopedia of Japan|Kodansha Encyclopedia of Japan]]. 1983. Supplement. 1986. [https://books.google.co.uk/books?id=WvApAQAAMAAJ]
*Dorothy Perkins. Encyclopedia of Japan: Japanese History and Culture, from Abacus to Zori. Facts on File. A Roundtable Press Book. 1991. [https://books.google.co.uk/books?id=JLKGAAAAIAAJ]
*Pictorial Encyclopedia of Modern Japan. Gakken. 1986. [https://books.google.co.uk/books?id=0FgKAQAAIAAJ]
*Boye Layfayette De Mente. Japan Encyclopedia. 1995. [https://books.google.co.uk/books?id=f9c7AAAAMAAJ]
**Boye De Mente. Everything Japanese. [The Authoritave Reference on Japan Today]. 1989. [https://books.google.co.uk/books?id=Duku89bARgoC]

Biographical dictionaries
*Wispelway. Japanese Biographical Index [https://books.google.co.uk/books?id=4UkhAAAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Seiichi Iwao. Biographical Dictionary of Japanese History. Kodansha International. 1978.[https://books.google.co.uk/books?id=TkQOAQAAMAAJ]
*Senʼichi Hisamatsu. Biographical Dictionary of Japanese Literature. Kodansha International. 1976. [https://books.google.com/books?id=X3lkAAAAMAAJ]
*Yutaka Tazawa (ed). Biographical Dictionary of Japanese Art. Kodansha International. 1981. [https://books.google.com/books?id=7ozrAAAAMAAJ]
*Japan Biographical Encyclopedia &amp; Who's Who. 1958. Review: [https://books.google.co.uk/books?id=lV_VAAAAMAAJ 25] Contemporary Japan 280
**Japan Biographical Encyclopedia &amp; Who's Who (2nd Edition) 1961. The Japan Biographical Research Department, The Rengo Press Ltd. Preface dated October 1960. Commentary: [https://books.google.co.uk/books?id=-SZSAAAAMAAJ]
***1961 Supplement. December 1961.
*Shunjiro Kurita. [[w:ja:Who's Who in Japan|Who's Who in Japan]]. [https://archive.org/details/WhosWhoInJapan1913/page/n24/mode/1up 2nd Ed]. Catalogue: [https://cir.nii.ac.jp/crid/1971712334741056302].
**[[w:ja:石川安次郎|Yasujiro Ishikawa]] (ed). Fifth Annual Edition. The Who's Who in Japan Publishing Department: The Keiseisha. No 15, 2-chome, Owari-cho, Ginza, Kyobashi-ku, Tokyo. Preface dated May 1916. [https://books.google.co.uk/books?id=joBJAAAAYAAJ]. Yasujiro Ishikawa's biography &quot;Ishikawa Yasujiro&quot; is on p 213.
*Who's Who in Japan 1984/85. International Culture Institute. Hong Kong. [https://books.google.co.uk/books?id=5a0MAQAAMAAJ]. Reviews: [https://books.google.co.uk/books?id=_xVnAAAAMAAJ 16(2)] World Affairs Report col 349; The Journal of the American Chamber of Commerce in Japan, [https://books.google.co.uk/books?id=Aa62AAAAIAAJ November 1984], p 71.
*Who's Who in Japanese Government. Review:
[https://books.google.co.uk/books?id=91I3AAAAIAAJ 29] RQ 304 (Winter 1989).

Reference books
*Nihon No Sanko Tosho. Volume 1: 1965. Volume 2: 1972.
**Guide to Japanese Reference Books. American Library Association. Chicago. 1966: [https://books.google.co.uk/books?id=0rflAAAAMAAJ]. Supplement. 1979: [https://books.google.co.uk/books?id=j05_F9OHzkQC]. Commentary: Encyclopedia of Library and Information Science, vol 21, [https://books.google.co.uk/books?id=H1pNvzr_n98C&amp;pg=PA149#v=onepage&amp;q&amp;f=false p 149].

Media
*[https://www.bbc.com/news/world-asia-pacific-15217593 Japan media guide]. News. BBC. 20 March 2023.
*Masaaki Kasagi. Mass Media in Japan. (Orientation seminars on Japan, number 14). 1983. [https://books.google.co.uk/books?id=odkgAAAAIAAJ]
*Routledge Handbook of Japanese Media [https://books.google.co.uk/books?id=zilKDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Publishers
*[https://www.publishersweekly.com/pw/by-topic/international/international-book-news/article/99729-get-to-know-these-japanese-publishing-companies.html Get to Know These Japanese Publishing Companies]. Publishers Weekly. 20 February 2026.

Press and journalism
*[https://reutersinstitute.politics.ox.ac.uk/digital-news-report/2025/japan Japan]. Reuters Institute for the Study of Journalism. 17 June 2025.
*Marjane Aalam and Philippe Régnier. The Japanese Press and Information System. The Graduate Institute of International Studies. Geneva. [https://books.google.co.uk/books?id=RTcbAQAAIAAJ]
*The Japanese Press: Past and Present. Japan Newspaper Publishers' and Editors' Association. [https://books.google.co.uk/books?id=5tcQAAAAIAAJ 1949].
*Anthony Rausch. Japanese Journalism and the Japanese Newspaper: A Supplemental Reader. [https://books.google.co.uk/books?id=mZrToQEACAAJ]
*Frank L Martin. The Journalism of Japan. 1918. [https://books.google.com/books?id=ruYzAQAAMAAJ]
*William De Lange. A History of Japanese Journalism. Japan Library. 1998. [https://books.google.co.uk/books?id=Rd5tb0cuz8QC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kanesada Hanazono. The Development of Japanese Journalism. Osaka. 1924. [https://books.google.co.uk/books?id=z99ZAAAAMAAJ]
*Kanesada Hanazono. Journalism in Japan and Its Early Pioneers. 1926. [https://books.google.co.uk/books?id=IGTFfLc4bq0C]
*César Castellvi. A Sociology of Journalism in Japan: The Last Empire of the Press. 2024. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PR4#v=onepage&amp;q&amp;f=false]
*&quot;Japan&quot;. Christopher H Sterling (ed). Encyclopedia of Journalism. A Sage Reference Publication. 2009. ISBN 9780761929574. vol 3. pp [https://books.google.co.uk/books?id=ZQhDq8fPj2IC&amp;pg=PA809#v=onepage&amp;q&amp;f=false 809] to 815.

Press annuals
*The Japanese Press. (Nihon Shinbun Kyokai). [https://books.google.co.uk/books?id=AfvyAAAAMAAJ 1979] [https://books.google.co.uk/books?id=Au3yAAAAMAAJ 1998]

Summaries of the press
*Daily Summary of Japanese Press

Foreign correspondents
*Foreign Correspondents in Japan: Reporting a Half Century of Upheavals, from 1945 to the Present. Tuttle. 1998. [https://books.google.co.uk/books?id=YI3TAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*G Raymond Nunn (comp). Japanese Periodicals and Newspapers in Western Languages: An International Union List. Mansell. 1979. [https://books.google.co.uk/books?id=jEROAQAAIAAJ]
*Japan Periodicals. Keizai Koho Center. 3rd Ed [https://books.google.co.uk/books?id=ATm0AAAAIAAJ]. Japan Periodicals, 1982. [https://books.google.co.uk/books?id=PkMyAAAAMAAJ]
*Japanese Periodicals Index
**Humanities and Social Sciences [https://books.google.co.uk/books?id=nXX_RpPGf3AC]
**Natural Sciences [https://books.google.co.uk/books?id=FCJIAAAAYAAJ]
*Current Japanese Periodicals [https://books.google.co.uk/books?id=FjO5AAAAIAAJ]
*Check-list of Japanese Periodicals Held in British University and Research Libraries. [https://books.google.co.uk/books?id=VZgsAAAAYAAJ]
*Union List of Current Japanese Periodicals in the East Asian Libraries of Columbia, Harvard, Princeton, and Yale Universities. [https://books.google.co.uk/books?id=yw7kAAAAMAAJ]
*List of Japanese Periodicals in the Library of the School of Oriental &amp; African Studies. [https://books.google.co.uk/books?id=RREjAQAAIAAJ]
*Gianni Simone. [https://www.japantimes.co.jp/community/2011/04/26/issues/english-mags-approach-milestone-crossroads/ English mags approach milestone, crossroads]. The Japan Times. 26 April 2011.
*Japan Report (1955 onwards) (Consulate General of Japan, Japan Information Center). Vol 39 published in 1993. [https://books.google.co.uk/books?id=MX4BN_frv4IC&amp;pg=PP7#v=onepage&amp;q&amp;f=false] editions:jYuMSMIQC-AC
**Japan Information
*Japan Now [https://books.google.co.uk/books?id=Nul7DRQaexMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Japan Quarterly. (Asahi Shimbun). 1954 to 2001. [https://books.google.co.uk/books?id=nZMMAQAAMAAJ] [https://books.google.co.uk/books?id=_RwVAAAAMAAJ] 189 issues.
*Japan Illustrated: The Japan Times Quarterly [Pictorial] Magazine (October 1963 to Summer 1977) 15 vols [https://books.google.co.uk/books?id=D7UThOmE8T4C]
*[[w:Japan Spotlight|Japan Spotlight]]. Economy, Culture &amp; History: Japan Spotlight: Bimonthly. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ]
*Focus Japan. (Japan External Trade Organization, JETRO). [https://books.google.co.uk/books?id=2fG2hsEZpRkC]
*The Japan Journal [https://books.google.co.uk/books?id=2V3hAAAAMAAJ] [https://books.google.co.uk/books?id=CJwoAQAAMAAJ]
*Japan Magazine. Muromachi Publicity Corporation. (vols 1 to 5: 1957 to 1963). [https://books.google.co.uk/books?id=Swd18PnVeUgC]
*The Japan Magazine: A Representative Monthly of Things Japanese [https://books.google.co.uk/books?id=ubGKo-p6O_0C] [https://archive.org/details/jm-1914-v4.9-5.2/mode/1up]
*Transactions and Proceedings of the Japan Society, London [https://books.google.co.uk/books?id=B75nnph5qHgC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
**Bulletin. [Bulletin of the Japan Society, London.] [https://books.google.co.uk/books?id=Pd9KvyhnpjMC]
**The Japan Society of London Bulletin [https://books.google.co.uk/books?id=XxlxAAAAMAAJ]
*About Japan. Japan Society, New York. [https://books.google.co.uk/books?id=Nf5OAQAAIAAJ]
**News Bulletin [https://archive.org/details/bub_gb_QcA3AQAAIAAJ/page/n2/mode/1up]
*[[w:en:Metropolis (free magazine)|Metropolis]] (metropolisjapan.com)
*[[w:en:Tokyo Weekender|Tokyo Weekender]] (トーキョー・ウィークエンダー) [https://www.tokyoweekender.com/japan-life/news-and-opinion/nhk-world-features-the-tokyo-weekender-magazine/]
*The Japan Gazette [https://books.google.co.uk/books?id=WSopAAAAYAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Tokio Times [https://books.google.co.uk/books?id=UDfiFBu0vB4C&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*[[w:en:Look Japan|Look Japan]]. (Look Japan Ltd). [https://books.google.co.uk/books?id=QnO6AAAAIAAJ]. Commentary: Gale Directory of Publications and Broadcast Media [https://books.google.co.uk/books?id=ve4dAQAAMAAJ]
*[[w:en:Japan Echo|Japan Echo]]. 1974 to 2010. [https://books.google.co.uk/books?id=Cmq6AAAAIAAJ] [https://books.google.co.uk/books?id=fpmEPpl-85UC]
*PHP Intersect. (Where Japan Meets Asia and the World). PHP Institute. [https://books.google.co.uk/books?id=i74TAQAAMAAJ]
**Intersect Japan [https://books.google.co.uk/books?id=sL8TAQAAMAAJ]
*Japan Digest [https://books.google.co.uk/books?id=8AcOAQAAMAAJ]
*Speaking of Japan [https://books.google.co.uk/books?id=U7S0AAAAIAAJ]. [Speeches.]
*The Hansei Zasshi: A Monthly Magazine [https://books.google.co.uk/books?id=6qBhfHZo7Q0C&amp;pg=PP5#v=onepage&amp;q&amp;f=false][https://books.google.co.uk/books?id=dyIsvnYjpwEC&amp;pg=PP6#v=onepage&amp;q&amp;f=false]
**The Orient. 1899 onwards [https://books.google.co.uk/books?id=nS1omYYnnd4C&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Today's Japan. Orient/West Incorporated. [https://books.google.co.uk/books?id=g2ASAAAAMAAJ]
*Japan Review: Bulletin of the International Research Center for Japanese Studies. [https://books.google.co.uk/books?id=GggOAQAAMAAJ]
*[https://www.bloomberg.com/account/newsletters/next-japan Next Japan]. Bloomberg.
*Niponica. [https://books.google.co.uk/books?id=X2scKgfQS5oC]

Newspapers

See also [[w:List of newspapers in Japan]]
*Haruhara Akihiko, &quot;English-language newspapers in Japan&quot; (1994) 41 Japan Quarterly [https://www.proquest.com/openview/8e2b760f2a2fa37ba164ea675c095353/1 474] (Issue 4: October 1994)
*James Charles Tanner. English Language Newspapers in Bakumatsu Japan. 1977. [https://books.google.co.uk/books?id=Bn2zgzZy_3oC]
*[https://www.japantimes.co.jp/news/2009/03/03/reference/newspapers-here-soldiering-on/ Newspapers here soldiering on]. The Japan Times. 3 March 2009.

*[[w:The Japan Times|The Japan Times]]
**The Japan Times: Weekly Edition [https://books.google.co.uk/books?id=KoQ-AQAAMAAJ] [https://books.google.co.uk/books?id=yYQ-AQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Japan Daily Mail
*Japan Weekly Mail
*The Japan Chronicle
**Weekly Edition [https://books.google.co.uk/books?id=vXdRAQAAIAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Japan News. (The Japan News by The Yomiuri Shimbun)
**Yomiuri Japan News (from 1955)
**The Yomiuri (from 1958)
**The Daily Yomiuri (from 1970)
*The Asahi Shimbun: Asia &amp; Japan Watch. [https://www.asahi.com/sp/ajw/]
**Asahi Evening News (from 1954)
***Tokyo Evening News (1952 to 1954) [https://ndlsearch.ndl.go.jp/books/R100000002-I000000145073]
*The Mainichi. [https://mainichi.jp/english/]
**Mainichi Daily News (1922 to 2001) [https://www.nytimes.com/2001/02/27/business/worldbusiness/IHT-tech-briefstop-the-presses.html] [https://ndlsearch.ndl.go.jp/books/R100000002-I000000144910]

Sports newspapers; sports dailies
*Louise do Rosario, &quot;News-stand stars&quot; in &quot;Japan&quot; (1992) [https://books.google.co.uk/books?id=T_GzAAAAIAAJ 155] [[w:en:Far Eastern Economic Review|Far Eastern Economic Review]], 24 to 31 December 1992, p 21
*[[w:ja:岡崎満義|Mitsuyoshi Okazaki]], &quot;Unsportsmanlike Journalism: Japan's sports dailies may be popular, but are they sporting?&quot; in &quot;Sport&quot;, [[w:en:Look Japan|Look Japan]], [https://books.google.co.uk/books?id=lD3tAAAAMAAJ January 1995], p 39

News
*[[w:en:Japan Today|Japan Today]] (ジャパントゥデイ). GPlusMedia. Gakken Holdings.

Annuals and year books 
*This is Japan. Asahi Shimbun. 1954 to 1971. [https://books.google.co.uk/books?id=2X9DAQAAIAAJ]. Commentary: A Victorian Sailor's Grave in the Seto Inland Sea, p 244 [https://books.google.co.uk/books?id=OegkAgAAQBAJ&amp;pg=PA244#v=onepage&amp;q&amp;f=false]
*The Japan Year Book. The Japan Year Book Office. 1905 onwards. [https://archive.org/details/bub_gb_arFPAAAAMAAJ/page/n10/mode/1up 1906]. [https://archive.org/details/in.ernet.dli.2015.553496/page/n27/mode/1up 1915].
*The &quot;Japan Gazette&quot; Japan Year Book. The Japan Gazette. [https://archive.org/details/japan-year-book-1913-1914/page/n15/mode/1up 1913-14]
*The Japan Times Year Book
*New Japan. Mainichi Newspapers. [https://books.google.co.uk/books?id=gRFCAQAAIAAJ]

Almanacs
*Asahi Shimbun Japan Almanac. [https://books.google.co.uk/books?id=SEEEAQAAIAAJ 1995].
*Japan Almanac. (The Mainichi Newspapers). [https://books.google.co.uk/books?id=ufAIAQAAIAAJ 1972]. [https://books.google.co.uk/books?id=X4eXWRkbtFsC 1973]. [https://books.google.co.uk/books?id=7rMrAAAAIAAJ] [https://books.google.co.uk/books?id=krMrAAAAIAAJ]
*[[w:Boyé Lafayette De Mente|Boye De Mente]]. Passport's Japan Almanac. [https://books.google.co.uk/books?id=741wAAAAMAAJ]

General
*Japan: A Country Study. (Area Handbook series). 4th Ed: 1983: [https://books.google.co.uk/books?id=HkM5N3JNc5IC]. 5th Ed: 1992: [https://books.google.co.uk/books?id=ze-wupXxpvEC]
*Area Handbook for Japan. 2nd Ed: 1964: [https://books.google.co.uk/books?id=WucdAAAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]. 3rd Ed: 1974: [https://books.google.co.uk/books?id=LG2aoq1U_eoC&amp;pg=PR1#v=onepage&amp;q&amp;f=false] (DA Pam 550-30).
*Colin Simpson. Picture of Japan.
**Japan: An Intimate View. A S Barnes. [https://books.google.co.uk/books?id=3hkeAAAAMAAJ]
**This is Japan. Angus &amp; Robertson. [https://books.google.co.uk/books?id=HJEJAQAAIAAJ]
*Japan. (The World and Its Peoples). Greystone Press, New York. 1964. Volume 1: [https://books.google.co.uk/books?id=yysUAQAAMAAJ]. Volume 2 &quot;Japan Korea&quot;, including Korea: [https://books.google.co.uk/books?id=uQAUAQAAMAAJ]. See pp 1 to 375 for Japan, and pp 376 to 379 for Ryukyu and Bonin Islands.
*Japan. (World and its Peoples: Eastern and Southern Asia, volume 8). Marshall Cavendish. 2008. ISBN 9780761476412.
*Edward Seidensticker. This Country, Japan. Kodansha International. 1979. ISBN 9780870112294. [https://books.google.co.uk/books?id=88wwAQAAIAAJ]
*Hall and Beardsley. Twelve Doors to Japan. McGraw-Hill. New York. 1965. [https://books.google.co.uk/books?id=0KpxAAAAMAAJ]
*&quot;Japan&quot;. Louis Barron (ed). &quot;Asia &amp; Australasia&quot;. Moshe Y Sachs (ed and pub). Worldmark Encyclopedia of the Nations. Worldmark Press, Inc. Harper &amp; Row New York. 1963. pp 145 to 160. [https://books.google.co.uk/books?id=I0oYAQAAMAAJ]
**&quot;Japan&quot;. Asia &amp; Oceania. 7th Ed: 1988. ISBN 0-471 62406-3. vol 4. pp 153 to 168.
**&quot;Japan&quot;. Asia &amp; Oceania. 8th Ed: 1995. Gale Research Inc. ISBN 0-8103-9882-6. vol 4. pp 203 to 220.
*James E Hoare. &quot;Japan&quot;. East and Southeast Asia. The World Today Series 2023–2024. pp [https://books.google.co.uk/books?id=3GfQEAAAQBAJ&amp;pg=PA107#v=onepage&amp;q&amp;f=false 107] to 133.
*Japan. (Library of Nations). Time Life Books. 1985. ISBN 0809453347. ISBN 0705408485. [https://books.google.co.uk/books?id=P1NJwcWAnboC] [https://books.google.co.uk/books?id=Ui2CvuyM4JkC]

Handbooks
*Patrick Heenan (ed). The Japan Handbook. (Regional Handbooks of Economic Development). 1998. [https://books.google.co.uk/books?id=IMG2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Introduction
*Introducing Japan Through Books: A Selected Bibliography. Public Information Bureau, Ministry of Foreign Affairs, Japan. 1968. [https://books.google.co.uk/books?id=FvsyAQAAIAAJ]. 2nd Ed: 1973: [https://books.google.co.uk/books?id=Vj0XAQAAMAAJ].
*Donald Ritchie. Introducing Japan. 1st Ed: 1978. Revised Ed: 1986. 6th printing: 1989: [https://books.google.co.uk/books?id=FE-nxxoKayQC]. 2nd Revised Ed: 1990. 2nd printing: 1991: [https://books.google.co.uk/books?id=hz4UAQAAIAAJ]. 1994: [https://books.google.co.uk/books?id=FMvT6m4SgIQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Herschel Webb. An Introduction to Japan. Columbia University Press. 2nd Ed: 1957: [https://books.google.co.uk/books?id=YQ8MAQAAIAAJ].
*Introducing Modern Japan. A publication of the Japan Information and Culture Center, Embassy of Japan.

Today and yesterday
*Ray Downs. Japan Yesterday and Today. Praeger Publishers. 1970. [https://books.google.co.uk/books?id=PwKxAAAAIAAJ]

Today
*Roger Buckley. Japan Today. Cambridge University Press. 3rd Ed [https://books.google.co.uk/books?id=thyqBtJp2DcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Contemporary
*Routledge Handbook of Contemporary Japan. 2021. [https://books.google.co.uk/books?id=yfH3DwAAQBAJ&amp;pg=PA2011#v=onepage&amp;q&amp;f=false]
*Duncan McCargo. Contemporary Japan. 3rd Ed: 2012. [https://books.google.co.uk/books?id=8I5KEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jeff Kingston. Contemporary Japan: History, Politics, and Social Change since the 1980s. 2011. [https://books.google.co.uk/books?id=enJQZA3R4FMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

[Series]
*Routledge Contemporary Japan Series

Modern
*Hugh Cortazzi. Modern Japan: A Concise Survey. Palgrave Macmillan. 1993. [https://books.google.co.uk/books?id=Cf--DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

The Japanese
*Peter Tasker. The Japanese: Portrait of a Nation. 1989 [https://books.google.com/books?id=Q1N8ld78wwQC]
**The Japanese: A Major Exploration of Modern Japan. [https://books.google.co.uk/books?id=CW-6AAAAIAAJ]
**Inside Japan: Wealth, Work and Power in the New Japanese Empire. 1987. [https://books.google.co.uk/books?id=2OJuAAAAMAAJ]
*Edwin O Reischauer. The Japanese. The Belknap Press of Harvard University Press. [https://books.google.co.uk/books?id=zrEqAAAAYAAJ]
*Edwin O Reischauer and Marius B Jansen. The Japanese Today: Change and Continuity. 1995. [https://books.google.co.uk/books?id=BTPNlLIy2soC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Japaneseness
*Ray T. Donahue (ed). Exploring Japaneseness: On Japanese Enactments of Culture and Consciousness. Ablex Publishing. 2002. [https://books.google.co.uk/books?id=V_NwAAAAMAAJ]
*Yoji Yamakuse. Japaneseness: A Guide to Values and Virtues. 2016. [https://books.google.co.uk/books?id=tB0iDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Travel books
*DK Eyewitness Travel: Japan. Reprinted with revisions. 2015: [https://books.google.co.uk/books?id=g2NaBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2017: [https://books.google.co.uk/books?id=vg15DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Dodd and Richmond. The Rough Guide to Japan. 2nd Ed: 2001: [https://books.google.co.uk/books?id=pRGq95ytWZoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Frommer's Japan. 5th Ed: 2000: [https://books.google.co.uk/books?id=-QC8mVyvPa8C].
*Fodor's Japan YYYY. 1984. [https://books.google.co.uk/books?id=aH2Ow27HUQ0C 1986]. [https://books.google.co.uk/books?id=3gTTf6nbv20C 1987]. 1988.
**Fodor's YY Japan. [https://books.google.co.uk/books?id=9QMHllzldlYC 91]. 92. 93.
**Fodor's Japan. 13th Ed: 1996: [https://books.google.co.uk/books?id=cZxZAAAAYAAJ]
*The New Official Guide: Japan. Japan Travel Bureau. 1966. [https://books.google.co.uk/books?id=HoxxAAAAMAAJ]
*Here is Japan. Asahi Broadcasting Corporation. [https://books.google.co.uk/books?id=8QXRCTMNG7MC]
*Japan. (Nagel Travel Guide Series, vol 32). 1964. [https://books.google.co.uk/books?id=QsbXAAAAMAAJ]
*Clark. All the Best in Japan: with Manila, Hong Kong, and Macao. (&quot;All the Best&quot; series). 1959. Reprinted 1964. [https://books.google.co.uk/books?id=yUq4YaaryrwC]. Reviews: [https://archive.dartmouthalumnimagazine.com/article/1958/6/1/all-the-best-in-japan] (1958) 110 Travel 51 [https://books.google.co.uk/books?id=UVwXAQAAMAAJ] 3 Bulletin of the Japan Society, London, No 11: June 1960, p 25 [https://books.google.co.uk/books?id=2oy74hRRXk4C]
**All the Best in Japan and the Orient. 1967.

Night
*[[w:ja:Taizo Fujimoto|Taizo Fujimoto]]. The Nightside of Japan. T Werner Laurie. Essex Street, Strand, London. 1914. [https://archive.org/details/nightsideofjapan00fujirich/page/n6/mode/1up]

Exhibitions
*Patricia Morison. &quot;Fun and games with Japanese angst&quot;. The Financial Times. Weekend 21/22 September 1991. No 31562. Section II: &quot;Weekend FT&quot;. Weekend 21/22 September 1991. Page xx. [&quot;Visions of Japan&quot; exhibition, Victoria and Albert Musuem, ending 5 January 1992.]

Languages

See [[Universal Bibliography/Languages/Japanese|Japanese]]

Literature

See [[Universal Bibliography/Literature#Japanese|Japanese literature]]

Music

See [[Universal Bibliography/Music#Japanese and Japan|Music of Japan]]

Cinema

See [[Universal Bibliography/Cinema#Japanese|Cinema of Japan]]

Television

See [[Universal Bibliography/Television#Japanese|Japanese television]]

Architecture

See [[Universal Bibliography/Architecture|Japanese architecture]]

Culture

See [[Universal Bibliography/Culture#Japanese|Culture of Japan]]

Economy
*MERI's Monthly Circular: Survey of Economic Conditions in Japan (1923 to 2025) Mitsubishi Economic Research Institute [https://books.google.co.uk/books?id=4QDtAAAAMAAJ] [https://www.meri.or.jp/en/mc.html]

Tokyo
*J S Eades. Tokyo. (World Bibliographical Series, vol 214). CLIO Press. 1999. ISBN 1-85109-292-7. [https://books.google.co.uk/books?id=TrkUAQAAIAAJ]

==Korea==
*Korea Journal [https://books.google.co.uk/books?id=O6XfBexsp6gC]

Bibliography and literature
*Thomas H Kang. &quot;Korean Literature and Bibliography&quot;. Kent, Lancour and Daily (eds). Encyclopedia of Library and Information Science. Marcel Dekker. 1977. vol 21. pp [https://books.google.co.uk/books?id=H1pNvzr_n98C&amp;pg=PA176#v=onepage&amp;q&amp;f=false 176] to 240.

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    <title>Universal Bibliography/Libraries</title>
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      <text bytes="11010" sha1="aa9i3d9gghb8dj4uk2uvpj9xcnsbltr" xml:space="preserve">{{Bibliography}}

This part of the [[Universal Bibliography]] is a bibliography of libraries and librarianship.

See [[s:Category:Libraries]] and [[:Category:Librarianship literature]]

==Metabibliography==
*Burton and Vosburgh. A Bibliography of Librarianship: Classified and Annotated Guide to the Library Literature of the World (excluding Slavonic and Oriental Languages). 1934. [https://trove.nla.gov.au/work/20629623]
*Thornton, John Leonard. Classics of Librarianship: Further Selected Readings in the History of Librarianship. Library Association. 1957. [https://books.google.co.uk/books?id=vXoYAAAAMAAJ]
*Programmed Instruction in Librarianship: A Classified Bibliography of Programmed Texts and Other Materials, 1960-1974 [https://books.google.co.uk/books?id=gvMs0gNS3MwC]

*Besterman. Bibliography, Library Science, and Reference Books: A Bibliography of Bibliographies. [https://books.google.co.uk/books?id=zKAZAAAAMAAJ]
*Bemis, Michael. Library and Information Science: A Guide to Key Literature and Sources. American Library Association. 2014. [https://books.google.co.uk/books?id=_9mgAwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*A Selected Bibliography of Library and Information Science. Center for Documentation and Communication Research, School of Library Science, Case Western Reserve University. Spring 1970. [https://books.google.co.uk/books?id=vqAaAAAAMAAJ]
*Bibliography of Library Science. 1970. [https://books.google.co.uk/books?id=pvEaAAAAMAAJ]
*T'u-shu kuan hsüeh lun-chu tsu-liao tsung-mu. A General Bibliography of Library Science. 1969. [https://books.google.co.uk/books?id=U3944PGx4U4C]
*Union List of Literature on Library Science and Bibliography Up to 1935. League of Young Librarians. 1938. [https://books.google.co.uk/books?id=E-vvAAAAMAAJ]

*Cannons. Bibliography of Library Economy. 1910 [https://archive.org/details/bibliographyofli00cann] [https://archive.org/details/bibliographyofli00cannrich]

*Bronson, Barbara. &quot;Bibliographical Guides to the History of American Libraries&quot;. University of Illinois Library School. Number 32. February 1953. [https://www.ideals.illinois.edu/bitstream/handle/2142/3961/gslisoccasionalpv00000i00032.pdf]

*Wynar. Research Methods in Library Science: A Bibliographic Guide with Topical Outlines. Libraries Unlimited. 1971. [https://books.google.co.uk/books?id=b_svAAAAYAAJ]
*Busha, Charles H. A Library Science Research Reader and Bibliographic Guide. Libraries Unlimited. 1981. [https://books.google.co.uk/books?id=4b9AAAAAMAAJ]

*Lewanski, Richard Casimir. Library Directories: Library Science Dictionaries. American Bibliographical Center. 1967. [https://books.google.co.uk/books?id=YVJCAAAAIAAJ]

*Karp, Rashelle S. The Academic Library of the 90s: An Annotated Bibliography. Greenwood Press. 1994. [https://books.google.co.uk/books?id=yq3gAAAAMAAJ]
*Lebovitch. College Libraries and Tutor Librarianship: An Annotated Select Bibliography. Hatfield Polytechnic. 1971. [https://books.google.co.uk/books?id=cSkEAAAAMAAJ]

*Liebscher, Peter (compiler). Audiovisual Librarianship: A Select Bibliography, 1965-1983. Aslib &amp; Library Association Audiovisual Groups. 1984. [https://books.google.co.uk/books?id=UbZaAAAAYAAJ]

*A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC]

*Local Studies Librarianship: A World Bibliography. Library Association. 2001. [https://books.google.co.uk/books?id=pqUVAQAAIAAJ]

*Nunn, Geoffrey Raymond. Asian Libraries and Librarianship: An Annotated Bibliography of Selected Books and Periodicals and a Draft Syllabus. Scarecrow Press. 1973. [https://books.google.co.uk/books?id=zwYZAAAAMAAJ]
*Pantelidis, Veronica S. The Arab World: Libraries and Librarianship, 1960-1976: A Bibliography. Mansell. 1979. [https://books.google.co.uk/books?id=EpCZAAAAIAAJ]

*Huq and Aman. Librarianship and the Third World: An Annotated Bibliography of Selected Literature on Developing Nations, 1960-1975. Garland Publishing.1977. [https://books.google.co.uk/books?id=uNTgAAAAMAAJ]

*A G Le Sueur. Bibliographic Guide to South African Librarianship. [https://books.google.co.uk/books?id=hapBAAAAMAAJ]
*Wei, Karen T. Library and Information Science in China: An Annotated Bibliography. Greenwood Press. 1988. [https://books.google.co.uk/books?id=QPfgAAAAMAAJ]

*Murtagh. Education for Librarianship in Africa: A Bibliography. 1968 [https://books.google.co.uk/books?id=z80YAAAAMAAJ]

*Schlachter and Thomson. Library Science Dissertations, 1973-1981: An Annotated Bibliography. Libraries Unlimited. 1982. [https://books.google.co.uk/books?id=tpUaAAAAMAAJ]
*Cohen, Denison and Boehlert. Library Science Dissertations: 1925-1960, An Annotated Bibliography of Doctoral Studies. Bureau of Educational Research and Development, Library Services Branch. Bulletin 1963, No 38. [https://books.google.co.uk/books?id=x-iGptgjHdoC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Kumar (compiler). Research in Library and Information Science in India. Concept Publishing Company. 1987. [https://books.google.co.uk/books?id=dZKz1_7FkEMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Bibliography of Theses, Submitted to the Department of Library Science, University of Karachi, 1973-75 [https://books.google.co.uk/books?id=tKoZAAAAMAAJ]

==Library associations==

See [[s:Portal:Library Association of the United Kingdom]]
*Monthly Notes of the Library Association of the United Kingdom [https://books.google.co.uk/books?id=iLcaAAAAMAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

==Glossaries==
*Harrod's Librarians' Glossary and Reference Book. [https://books.google.co.uk/books?id=aOEFDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Rare books==
Rare book librarianship
*Pearson, &quot;Rare book librarianship and historical bibliography&quot; in Bowman (ed). British Librarianship and Information Work 1991-2000. Ashgate. 2006. Chapter 10. Pages [https://books.google.co.uk/books?id=yKousnrk2hEC&amp;pg=PA168#v=onepage&amp;q&amp;f=false 168] to 182.
*Galbraith and Smith. Rare Book Librarianship: An Introduction and Guide. Libraries Unlimited. 2012. [https://books.google.co.uk/books?id=NxkFdaRnuqkC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Cave Roderick. Rare Book Librarianship. Clive Bingley. 1982. [https://books.google.co.uk/books?id=GfDgAAAAMAAJ]
*A Rare Book Manual. Association of College and Reference Libraries, Rare Book Section. 1959. [https://books.google.co.uk/books?id=Af60vE3veWsC]

*Bennett, Josiah Q. The Cataloguing Requirements of the Book Division of a Rare Book Library. Kent State University Libraries. 1969. [https://books.google.co.uk/books?id=ICa6AAAAIAAJ]

*McCrank, Lawrence J. Education for Rare Book Librarianship: A Reexamination of Trends and Problems. (Occasional Papers No 144). University of Illinois , Graduate School of Library Science. April 1980.

*Selected Reference Works for Rare Book Librarians Exhibited in the National Library of Medicine. [https://books.google.co.uk/books?id=BM9NAQAAIAAJ&amp;pg=PP3#v=onepage&amp;q&amp;f=false]

Periodicals:
*Rare Books and Manuscripts Librarianship. (Association of College and Research Libraries). [https://books.google.co.uk/books?id=ROBMAQAAIAAJ]

Rare book libraries, rooms and collections
*Wendorf, Richard (ed). Rare Book and Manuscript Libraries in the Twenty-first Century. Harvard University Press. 1993. (1993) [https://books.google.co.uk/books?id=5RjhAAAAMAAJ 4] Harvard Library Bulletin (New Series), Numbers 1 and 2.

*Archer, Horace Richard. Rare Book Collections: Some Theoretical and Practical Suggestions for Use by Librarians and Students. 1965: [https://books.google.co.uk/books?id=MjDpAAAAMAAJ]. 1981.

*Attar, Karen. Directory of Rare Book and Special Collections in the UK and Republic of Ireland. 3rd Ed: 2016: [https://books.google.co.uk/books?id=sQUUDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Bloomfield, Barry Cambray (ed). 1997: [https://books.google.co.uk/books?id=1Y4UAQAAIAAJ]
*The Rare Books Division: A Guide to Its Collections and Services. The Library of Congress. 1950. [https://books.google.co.uk/books?id=sQvAAAAAIAAJ]
*Boneham, John. Rare Book Collections in Higher Education Libraries in Wales. 2013. [https://books.google.co.uk/books?id=kdUUyAEACAAJ]
*The Thomas Fisher Rare Book Library: A Brief Guide to the Collections. University of Toronto. 1982. [https://books.google.co.uk/books?id=4e9pAAAAMAAJ]
*Rare Book Collection in the School of Law Library, Southern Illinois University at Carbondale. 1982. [https://books.google.co.uk/books?id=hLHxRjklqcMC]
*Hackett, Joan. The Rare Book Collection in the Liberal Arts College Library. 1965. [https://books.google.co.uk/books?id=OKYZzQEACAAJ]

*Bishop, &quot;Rare Book Rooms in Libraries&quot; (1942) Library Quarterly [https://books.google.co.uk/books?id=BcYJAAAAIAAJ]

*An Examination of the Publications of Some Rare Book Libraries and Collections, 1952-1956. [https://books.google.co.uk/books?id=rlzpAAAAMAAJ]

*Overmier, Judith Ann. Scientific Rare Book Collections in Academic and Research Libraries in Twentieth Century America. University of Minnesota. 1985. [https://books.google.co.uk/books?id=BszUZbM53QIC]

Dictionaries and thesauruses:
*Dictionary Catalog of the Rare Book Division. [https://books.google.co.uk/books?id=ytpOAQAAMAAJ]
*Thesauri for Use in Rare Book and Special Collections Cataloguing Series 
**Genre Terms: A Thesaurus for Use in Rare Book and Special Collections Cataloguing. [https://books.google.co.uk/books?id=Q9PQhknsKSQC]
**Binding Terms: A Thesaurus for Use in Rare Book and Special Collections Cataloging. [https://books.google.co.uk/books?id=SQGx7s3he9wC]
**Type Evidence: A Thesaurus for Use in Rare Book and Special Collections Cataloguing. [https://books.google.co.uk/books?id=V_hAiCr7t2oC]

Cataloging
*Descriptive Cataloging of Rare Books. Library of Congress. 2nd Ed: 1991. [https://books.google.co.uk/books?id=4zikZtosctwC]
*Examples to Accompany Descriptive Cataloging of Rare Books. Association of College and Research Libraries. 1993. 1999. [https://books.google.co.uk/books?id=RxIDmedwCnsC]
*Lennon and Pearson. Rare Book Cataloguing in the British Isles. 1991. [https://books.google.co.uk/books?id=IHksAAAAYAAJ]

Catalogs
*The Imprint Catalog in the Rare Book Division [https://books.google.co.uk/books?id=ODhKAQAAMAAJ]
*Library of Congress Acquisitions: Rare Book and Special Collections Division [https://books.google.co.uk/books?id=RNagR4YKxOMC]

==Security==
*Wilkie. Guide to Security Considerations and Practices for Rare Book, Manuscript, and Special Collections Libraries. 2011. [https://books.google.co.uk/books?id=_ecESDdC6DQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

==Periodicals==

See [[Universal Bibliography/Periodicals#Libraries|Periodicals, Libraries]]

*Herald of Library Science [https://books.google.co.uk/books?id=ztPgAAAAMAAJ]

==Librarianship as a career==
*Tomlinson, Norman. Librarianship. (Target for Careers). Robert Hale. 1964. [https://books.google.co.uk/books?id=qDQOwp_r1rQC]

==Arizona==
*Arizona Librarian
*Arizona Library News
*Carole Marsh, The Arizona Library Book, 1991, 1994
*Libraries of the University of Arizona, 1977

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      <comment>Removing [[:c:File:Baylonianmaps.JPG|Baylonianmaps.JPG]], it has been deleted from Commons by [[:c:User:Ruthven|Ruthven]] because: per [[:c:Commons:Deletion requests/File:Baylonianmaps.JPG|]].</comment>
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      <text bytes="129932" sha1="7734ylnichu7kf24i0ba75ketngot8v" xml:space="preserve">[[Image:Gurue Mount Murresse.jpg|thumb|right|250px|This diversified and partly man-shaped landscape in northern Mozambique has some visual appeal (being ''beautiful'' or ''picturesque''). Credit: [[w:User:Paulo Oliveira|Paulo Oliveira]].]]
'''Archaeology''' &quot;studies human cultures through the recovery, documentation and analysis of material remains and environmental data, including architecture, artifacts, ecofacts, human remains, and landscapes.&quot;&lt;ref name=Crazedandinfused&gt;{{ cite book
|author=Crazedandinfused
|title=Topic:Archeology
|date=September 6, 2007
|url=http://en.wikiversity.org/w/index.php?title=Topic:Archeology&amp;diff=156532&amp;oldid=155948
|accessdate=2013-01-13 }}&lt;/ref&gt;

It is the study of [[w:human|human]] activity in the past, primarily through the recovery and [[w:analysis|analysis]] of the [[w:material culture|material culture]] and environmental data that they have left behind, which includes [[w:Artifact (archaeology)|artifacts]], [[w:architecture|architecture]], [[w:biofact (archaeology)|biofact]]s and [[w:cultural landscapes|cultural landscapes]] (the [[w:archaeological record|archaeological record]]).

Because archaeology employs a wide range of different procedures, it can be considered to be both a [[science]] and a [[Draft:humanities|humanity]].&lt;ref name=&quot;Renfrew_Bahn1991&quot;&gt;Renfrew and Bahn (2004 [1991]:13)&lt;/ref&gt;

Archaeology studies human [[history]] from the development of the first [[w:stone tools|stone tools]] in eastern Africa 3.4 million years ago up until recent decades.&lt;ref name=&quot;consumption&quot;&gt;McPherron, S. P., Z. Alemseged, C. W. Marean, J. G. Wynn, D. Reed, D. Geraads, R. Bobe, and H. A. Bearat. 2010. Evidence for stone-tool-assisted consumption of animal tissues before 3.39 million years ago at Dikika, Ethiopia. Nature 466:857-860&lt;/ref&gt; (Archaeology does not include the discipline of [[Draft:Paleontology|paleontology]].) It is of most importance for learning about [[w:Prehistory|prehistoric societies]], when there are no written records for historians to study, making up over 99% of total human history, from the [[w:Palaeolithic|Palaeolithic]] until the advent of literacy in any given society.&lt;ref name=&quot;Renfrew_Bahn1991&quot;/&gt;
{{clear}}

==Theoretical archaeology==
[[Image:Archaeological Artifact.jpg|right|thumb|150px|An old lamp was used in houses. Credit: [[c:user:Reniths|Renith Sunil.]]{{tlx|free media}}]]
'''Def.''' the &quot;period of time that has already happened&quot;&lt;ref name=PastWikt&gt;{{ cite book
|author=[[wikt:User:Tormod|Tormod]]
|title=past
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=13 July 2004
|url=http://en.wiktionary.org/wiki/past
|accessdate=2015-03-28 }}&lt;/ref&gt; is called the '''past'''.

'''Def.''' the &quot;[[wikt:study|study]] of the past through [[wikt:material|material]] [[wikt:remains|remains]]&quot;&lt;ref name=ArchaeologyWikt&gt;{{ cite book
|title=archaeology
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=October 16, 2012
|url=http://en.wiktionary.org/wiki/archaeology
|accessdate=2012-12-30 }}&lt;/ref&gt; is called '''archaeology''', or '''archeology'''.
{{clear}}

==Entities==
[[Image:Gordon Childe.jpg|thumb|right|250px|This is a photograph taken of the noted archaeologist and socialist V. Gordon Childe, circa 1930s. Credit: Swan Watson, Andrew.]]
&quot;[D]ominant groups create and control the meanings and uses of material culture. If other groups wish to be understood by the dominant group, they must express themselves through the goods controlled by the dominant group.&quot;&lt;ref name=Shackel&gt;{{ cite book
|author=Paul A. Shackel
|title=Craft to wage labor Agency and resistance in American historical archaeology, In: ''Agency in Archaeology''
|publisher=Routledge
|location=London
|date=2000
|editor=Marcia-Anne Dobres and John E. Robb
|pages=
|url=http://www.bsos.umd.edu/ANTH/People/FacStaff/faculty/pshackel/Publications/Craft%20to%20Wage%20Labor,%20Agency%20and%20Resistance%20in%20American%20Historical%20Archaeology%202000.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=
|accessdate=2011-12-06 }}&lt;/ref&gt;

&quot;But, the privilege afforded a certain dominant group of 'normal' archaeologists in terms of their ways of constructing the past influences all aspects of archaeological practice.&quot;&lt;ref name=Dowson&gt;{{ cite journal
|author=Thomas A. Dowson
|title=Why queer archaeology? An introduction
|journal=World Archaeology
|year=2000
|volume=32
|issue=2
|pages=161-5
|url=http://www.tandfonline.com/doi/abs/10.1080/00438240050131144
|arxiv=
|bibcode=
|doi=10.1080/00438240050131144
|pmid=
|accessdate=2012-01-03 }}&lt;/ref&gt;

&quot;However, some answers to these questions may emerge from a consideration of the dominant group, that is, the five institutions whose scholars published the most articles in the two periodicals.&quot;&lt;ref name=Dyson&gt;{{ cite journal
|author=Stephen L. Dyson
|title=Two Paths to the past: A Comparative Study of the Last Fifty Years of American Antiquity and the American Journal of Archaeology
|journal=American Antiquity
|month=April
|year=1985
|volume=50
|issue=2
|pages=452-63
|url=http://www.jstor.org/stable/280503
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2012-01-03 }}&lt;/ref&gt;
{{clear}}

==Landscape archaeology==
[[Image:Rural landscape - geograph.org.uk - 440894.jpg|right|thumb|300px|Rural landscape: A steep field is viewed from the B7068 with more pasture beyond by the Water of Milk. Credit: [https://www.geograph.org.uk/profile/6638 Walter Baxter].{{tlx|free media}}]]
'''Def.''' an assemblage of surfaces that are a portion of land, region, or territory, observable in its entirety is called a '''landscape'''.

'''Landscape'''  comprises the visible features of an area of [terrestrial ecoregion] land, including the physical elements of [[w:landform|landform]]s such as (ice-capped) mountains, hills, water bodies such as rivers, lakes, ponds and the sea, living elements of [[w:land cover|land cover]] including indigenous [[w:vegetation|vegetation]], human elements including different forms of [[w:land use|land use]], buildings and structures, and transitory elements such as [[w:lighting|lighting]] and [[w:weather|weather]] conditions.

'''Landscape archaeology''' is the study of the ways in which people in the past constructed and used the environment around them. Landscape archaeology is inherently multidisciplinary in its approach to the study of culture, and is used by both pre-historical, classic, and historic archaeologists. The key feature that distinguishes landscape archaeology from other archaeological approaches to sites is that there is an explicit emphasis on the study of the relationships between material culture, human alteration of land/cultural modifications to landscape, and the natural environment.
{{clear}}

==Sources==
[[Image:Dolina-Pano-3.jpg|thumb|right|250px|Excavations at the site of Gran Dolina, in Atapuerca (Spain), during 2008, are shown. Credit: [[commons:User:Mario Modesto Mata|Mario Modesto Mata]].]]
'''Archaeological field survey''' is the method by which archaeologists (often landscape archaeologists) search for archaeological sites and collect information about the location, distribution and organization of past human cultures across a large area (e.g. typically in excess of one hectare, and often in excess of many km&lt;sup&gt;2&lt;/sup&gt;).

Surveys are conducted to search for particular archaeological sites or kinds of sites, to detect patterns in the distribution of material culture over regions, to make generalizations or test hypotheses about past cultures, and to assess the risks that development projects will have adverse impacts on archaeological heritage.&lt;ref name=&quot;Banning 2002&quot;&gt;{{ cite book
|author=E. B. Banning
|title=Archaeological Survey
|date=2002
|publisher=Kluwer Academic Press
|location=New York }}&lt;/ref&gt;

The surveys may be: (a) ''intrusive'' or ''non-intrusive'', depending on the needs of the survey team (and the risk of destroying archaeological evidence if intrusive methods are used) and; (b) ''extensive'' or ''intensive'', depending on the types of research questions being asked of the landscape in question. Surveys can be a practical way to decide whether or not to carry out an excavation (as a way of recording the basic details of a possible site), but may also be ends in themselves, as they produce important information about past human activities in a regional context.

Excavation is the exposure, processing and recording of archaeological remains. An excavation site or &quot;dig&quot; is a site being studied.
{{clear}}

==Objects==
{{main|Rocks/Rocky objects}}
[[Image:Stone sphere.jpg|right|thumb|300px|A stone sphere created by the Diquís culture is in the courtyard of the Museo Nacional de Costa Rica (National Museum of Costa Rica). Credit: [[w:user:WAvegetarian|WAvegetarian]].{{tlx|free media}}]]
[[Image:Farm 6 archaeological site, Costa Rica.jpg|thumb|left|250px|Here is a view of the Farm 6 Archaeological site. Credit: A. Egitto.]]
[[Image:Pre-Columbian Stone Balls at Palmar Sur, Costa Rica.jpg|thumb|center|250px|Palmar Sur airport park has stone spheres. Credit: [[c:user:Matthewobrien|Matthewobrien]].]]
[[Image:Dropa disc.jpg|right|thumb|300px|A small disc with tiny hieroglyphics written on it apparently telling the story of an ancient race of aliens called the Dropa. Credit: Evelyn H. Armstrong.{{tlx|fairuse}}]]
'''Def.''' an &quot;object, such as a tool, weapon or ornament, [ceramics or pottery,] structure or finding in an experiment or investigation ... made or shaped by some agent or intelligence, ... [as] a result of external action, the test arrangement, or an experimental error ... rather than an [[wikt:inherent|inherent]] element&quot;&lt;ref name=ArtifactWikt&gt;{{ cite book
|title=artifact
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=12 December 2014
|url=http://en.wiktionary.org/wiki/artifact
|accessdate=2015-03-28 }}&lt;/ref&gt; is called an '''artifact''', or '''artefact'''.

In June 2014, the Precolumbian Chiefdom Settlements with Stone Spheres of the Diquis was added to the UNESCO list of World Heritage Sites.&lt;ref&gt;{{cite book|url=http://whc.unesco.org/en/news/1160|title=Six new sites inscribed on World Heritage List|publisher=UNESCO|accessdate=23 June 2014 }}&lt;/ref&gt;

The spheres range in size from a few centimetres to over {{convert|2|m|ft}} in diameter, and weigh up to 15 tons.&lt;ref name=&quot;bbc2010&quot;&gt;{{cite book |title=The stone spheres of Costa Rica |url=http://news.bbc.co.uk/1/hi/sci/tech/8593717.stm |date=29 March 2010 |publisher=BBC News |accessdate=2010-03-31}}&lt;/ref&gt;  Most are sculpted from gabbro.&lt;ref name=&quot;bbc2010&quot;/&gt;

The culture of the people who made them disappeared after the Spanish conquest.&lt;ref name=&quot;lynch&quot;&gt;{{cite book|url=http://www.eurekalert.org/pub_releases/2010-03/uok-uok032210.php|title=University of Kansas researcher investigates mysterious stone spheres in Costa Rica|author=Brendan M. Lynch|date=22 Mar 2010|accessdate=2010-03-24}}&lt;/ref&gt;

The first scientific investigation of the spheres was undertaken shortly after their discovery and published in 1943 in ''American Antiquity'', attracting the attention of Samuel Kirkland Lothrop&lt;ref&gt;{{cite book| title=Samuel Kirkland Lothrup, In: ''Biographical memoirs'', Volume 48| url= https://books.google.com/books?id=43U3AD_F9UMC&amp;pg=PA253| page=253| author=National Academy of Sciences| publisher=National Academies Press | date=1877| accessdate=2010-03-31 }}&lt;/ref&gt; of the Peabody Museum of Archaeology and Ethnology at Harvard University.&lt;ref&gt;{{cite book| author=Tim McGuinness| title=Costa Rican Diquis Spheres: Sphere history| publisher=mysteryspheres.com| accessdate=2010-03-31| url=https://web.archive.org/web/20100329102018/http://www.mysteryspheres.com/history.htm}}&lt;/ref&gt; In 1948, he and his wife attempted to excavate an unrelated archaeological site in the northern region of Costa Rica.&lt;ref&gt;{{cite book| title= Prehistoric Stone Balls—a Mystery, In: ''Picks from the Past''| url= http://naturalhistorymag.com/print/1353| author=Eleanor Lothrop| publisher=Natural History | date=September 1955| accessdate=2010-03-31}}&lt;/ref&gt; In San José he met Doris Stone, who directed the group toward the Diquís Delta region in the southwest (&quot;Valle de Diquís&quot; refers to the valley of the lower Río Grande de Térraba, including the Osa Canton towns of Puerto Cortés, Palmar Norte, and Sierpe.&lt;ref&gt;[http://www.mobot.org/MOBOT/Research/costaricagaz.shtml#Diquis Gazetteer of Costa Rican Plant-Collecting Locales: Diquís (or Dikís)] from the website of the Missouri Botanical Garden&lt;/ref&gt;

The Dropa stones or discs may be a series of at least 716 circular stone discs, dating back 12,000 years, on which tiny hieroglyph-like markings may be found.&lt;ref name=&quot;Fanthorpe&quot;&gt;{{cite book|author1=R. Lionel Fanthorpe|author2=Patricia Fanthorpe|author3=P. A. Fanthorpe|title=Mysteries and Secrets of the Masons: The Story Behind the Masonic Order|url=https://books.google.com/books?id=MXIIDlFKgcEC&amp;pg=PA40&amp;dq=dropa+stones|year=2006|publisher=Dundurn Press|isbn=978-1-55002-622-1|location=Toronto, Canada|pages=39–41}}&lt;/ref&gt;&lt;ref name=&quot;Vintner2011&quot;&gt;{{cite book|author=J. C. Vintner|title=Ancient Earth Mysteries|url=https://books.google.com/books?id=BzhZ_C3ccbwC&amp;pg=PA23|date=2 September 2011|publisher=AEM Publishing|page=23 }}&lt;/ref&gt; Each disc is claimed to measure up to {{convert|1|ft|cm}} in diameter and carry two grooves, originating from a hole in their center, in the form of a double spiral.&lt;ref name=BadArchaeology&gt;{{cite book |title=The Dropa (or Dzopa) stones |url=http://www.badarchaeology.com/out-of-place-artefacts/mysterious-objects/the-dropa-or-dzopa-stones/ |publisher=Bad Archaeology|date=7 May 2007 | author=Keith Fitzpatrick-Matthews |accessdate=12 February 2012}}&lt;/ref&gt;
{{clear}}

==Coppers==
[[Image:Ancient copper pot.JPG|right|thumb|300px|Ancient copper pot on three iron feet with handle is 15cm high. Credit: [[c:user:JaneArt|JaneArt]].{{tlx|free media}}]]
In &quot;Serbia [are] the oldest copper implements in the world. Some 7000 years ago, tradesmen of great wealth flourished on the Balkan peninsula.&quot;&lt;ref name=Franz&gt;{{ cite book
|author=Angelika Franz, translated from the German by Ami de Grazia
|title=Balkans: Archaeologists puzzle over 7,000y old copper-find - a tremendous fire destroyed a flourishing city
|publisher=Der Spiegel
|location=
|date=27 December 2010
|url=http://www.q-mag.org/balkans-archaeologists-puzzle-over-7000y-old-copper-find-a-tremendous-fire-destroyed-a-flourishing-city.html
|accessdate=2014-10-22 }}&lt;/ref&gt;

&quot;The soil around Plonik is rich in copper - the metal in a pure state lays often directly at the surface.&quot;&lt;ref name=Krauss&gt;{{ cite book
|author=Raiko Krauss, translated from the German by Ami de Grazia
|title=Balkans: Archaeologists puzzle over 7,000y old copper-find - a tremendous fire destroyed a flourishing city
|publisher=Der Spiegel
|location=
|date=27 December 2010
|url=http://www.q-mag.org/balkans-archaeologists-puzzle-over-7000y-old-copper-find-a-tremendous-fire-destroyed-a-flourishing-city.html
|accessdate=2014-10-22 }}&lt;/ref&gt;

The so-called &quot;copper-violets&quot; &quot;are related to the Alpine violets, which we know in Germany from flower pots. But they grow only in a soil with a very high copper content. There - where the soil is too poisonous for the other plants, they blossom in wide carpets - and in so doing point to the presence of copper.&quot;&lt;ref name=Krauss/&gt;

In &quot;the late stone age people picked up the beautiful stones. For instance, malachite. The green-hued mineral belongs to the class of carbonates. Copper content: 57 percent. Our ancestors made experiments with different stones over the fire. From there, it was only a small step to jewels and to more practical objects.&quot;&lt;ref name=Franz/&gt;

On &quot;the Black Sea Coast in today's Bulgaria, there flourished a prosperous, big city, quite close to today's Varna. Archaeologists have found the cemetary of this settlement. The graves were loaded with treasures: jewels - mostly gold, but also copper - sea shells from the Aegean and tools made of types of stone which must have come from far away.&quot;&lt;ref name=Franz/&gt;

&quot;In the 5. Millennium BC people lived there in complex, centralized settlements of up to a thousand inhabitants. They invented a new ceramic oven with two chambers which was remarkably well suited to the extraction of metal. We must strongly surmise that there existed relations between the Balkans and Northern Mesopotamia.&quot;&lt;ref name=Franz/&gt;

&quot;An enormous fire unleashed itself around 5000 BC in the city [of Plonik]. The earth carbonized. The houses collapsed and buried all the possessions of the inhabitants under their walls. When the flame died down, the city had ceased to exist.&quot;&lt;ref name=Franz/&gt;
{{clear}}

==Tools==
[[Image:Ancient tools in Khaplu.jpg|right|thumb|300px|This is a photo of a monument in Pakistan identified as the
GB-7. Credit: [[c:user:Aamirsohail56|Aamirsohail56]].{{tlx|free media}}]]
In &quot;the deep Southeast of Europe, near today's village of Plonik in Serbia, there existed 7000 years ago a major city. Its inhabitants lived in closely assembled huts and they melted copper to make jewels, tools and weapons.&quot;&lt;ref name=Franz/&gt;

&quot;The age of the oldest pieces which they have found up to now in the settlement is up to 7300 years. That's a good 800 years older as any other copper implements which have been found anywhere on Earth to this day.&quot;&lt;ref name=Franz/&gt;
{{clear}}

==Weapons==
[[Image:Neizeh metropolitan.JPG|right|thumb|200px|Iranian shafted weapons, 19th century, are from the Qajar era and are steel, damascened with gold. Credit: [[c:user:Zereshk|Zereshk]].{{tlx|free media}}]]
&quot;The people of the Balkan peninsula had a good knowledge of how they could create jewels, tools and weapons out of earth with a high copper content.&quot;&lt;ref name=Franz/&gt;

&quot;In Plonik, excavators found hatchets of copper. And weapons: axes and maces. With them a new area began. The stone age was gone, the copper age - chalcolithic - had begun.&quot;&lt;ref name=Franz/&gt;
{{clear}}

==Jewelry==
[[Image:Ancient Gold Jewelry (5986572251).jpg|right|thumb|300px|Ancient gold jewelry is displayed. Credit: [https://www.flickr.com/people/62091376@N03 Erik Drost].{{tlx|free media}}]]
&quot;From one grave, the excavator dug 1,516 K of gold jewelry. This is more gold than has been found in all the rest of the world for this particular epoch.&quot;&lt;ref name=Franz/&gt;
{{clear}}

==Materials==
{{main|Chemicals/Materials}}
[[Image:Landesmuseum Württemberg Stuttgart Antike 052.jpg|right|thumb|300px|Ancient ceramic statues are on display. Credit: [[c:user:Einsamer Schütze|Einsamer Schütze]].{{tlx|free media}}]]
&quot;In order to interpret archaeological ceramic assemblages in terms of social identities, a method was developed [that] consists in sorting the potsherds according to, successively, technical, techno-petrographic and morpho-stylistic criteria.&quot;&lt;ref name=Roux&gt;{{ cite book
|author=Valentine Roux and Marie-Agnès Courty
|title=Identifying social entities at a macro-regional level: Chalcolithic ceramics of South Levant as a case study, In: ''Pottery Manufacturing Processes''
|volume=
|publisher=University of Paris
|location=
|date=2005
|editor=
|pages=201-14
|url=http://www.mae.u-paris10.fr/prehistoire/IMG/pdf/RouxCourtyBAR2005.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=
|accessdate=2014-05-13 }}&lt;/ref&gt;
{{clear}}

==Heritage management==
[[Image:Cultural heritage management, Lapataia (8579148617).jpg|right|thumb|300px|Birgitte Skar are walking the trail in Bahia Lapataia, Parque Nacional, Tierra del Fuego, in order to learn how cultural heritage is defined, assessed and activated in contemporary cultural heritage management in Argentina. Credit: Photo K. Ø. Oftedal, [https://www.flickr.com/people/38254448@N05 NTNU Vitenskapsmuseet].{{tlx|free media}}]]
&quot;Rare, though, is the country where ethnic groups balance each other in terms of numbers, wealth or political influence and, consequently, it is not uncommon for the dominant group to use its power to push its own heritage to the fore, minimizing or denying the significance of subordinate groups as it crafts a national identity in its own image.&quot;&lt;ref name=Byrne&gt;{{ cite journal
|author=Denis Byrne
|title=Western hegemony in archaeological heritage management
|journal=History and Anthropology
|year=1991
|volume=5
|issue=2
|pages=269-76
|url=http://www.tandfonline.com/doi/abs/10.1080/02757206.1991.9960815
|arxiv=
|bibcode=
|doi=10.1080/02757206.1991.9960815
|pmid=
|accessdate=2012-02-16 }}&lt;/ref&gt;
{{clear}}

==Historical archaeology==
[[Image:Historical archaeology at the old Champoeg townsite, Champoeg State Park, Oregon (USA), 1973 (2144044691).jpg|right|thumb|200px|Photo was taken in the summer of 1973. Credit: [https://www.flickr.com/people/72105154@N00 John Atherton].{{tlx|free media}}]]
Historical archaeology is a form of archaeology dealing with places, things, and issues from the past or present when written records and oral traditions can inform and contextualize cultural material. These records can both complement and conflict with the archaeological evidence found at a particular site. Studies focus on literate, historical-period societies as opposed to non-literate, prehistoric societies. While they may not have generated the records, the lives of people for whom there was little need for written records, such as the working class, slaves, indentured labourers, and children but who live in the historical period can also be the subject of study. The sites are found on land and underwater. Industrial archaeology, unless practiced at industrial sites from the prehistoric era, is a form of historical archaeology concentrating on the remains and products of industry and the Industrial era.
{{clear}}

==Maya civilization==
[[Image:Tikal Jungle Structures.jpg|right|thumb|300px|Dense forest surrounds the city center of Tikal. Credit: Marcello A. Canuto, Francisco Estrada-Belli, Thomas G. Garrison, Stephen D. Houston, Mary Jane Acuña, Milan Kováč, Damien Marken, Philippe Nondédéo, Luke Auld-Thomas, Cyril Castanet, David Chatelain, Carlos R. Chiriboga, Tomáš Drápela, Tibor Lieskovský, Alexandre Tokovinine, Antolín Velasquez, Juan C. Fernández-Díaz, Ramesh Shrestha.{{tlx|fairuse}}]]
[[Image:Guatemala 074.jpg|thumb|right|300px|This is an image of the Mayan temple at Tikal in Guatemala. Credit: [[:nl:user:Rialfver|Rialfver]].{{tlx|free media}}]]
In the image on the right, dense forest surrounds the city center of this Classic-era Maya site (top) Tikal. Laser mapping of the same view (bottom) reveals structures and causeways hidden by the jungle.

&quot;Lidar (a type of airborne laser scanning) provides a powerful technique for three-dimensional mapping of topographic features.&quot;&lt;ref name=Canuto&gt;{{ cite journal
|author=Marcello A. Canuto, Francisco Estrada-Belli, Thomas G. Garrison, Stephen D. Houston, Mary Jane Acuña, Milan Kováč, Damien Marken, Philippe Nondédéo, Luke Auld-Thomas, Cyril Castanet, David Chatelain, Carlos R. Chiriboga, Tomáš Drápela, Tibor Lieskovský, Alexandre Tokovinine, Antolín Velasquez, Juan C. Fernández-Díaz, Ramesh Shrestha 
|title=Ancient lowland Maya complexity as revealed by airborne laser scanning of northern Guatemala
|journal=Science
|date=28 September 2018
|volume=361
|issue=6409
|pages=1355
|url=http://science.sciencemag.org/content/361/6409/eaau0137
|arxiv=
|bibcode=
|doi=10.1126/science.aau0137
|pmid=
|accessdate=7 October 2018 }}&lt;/ref&gt;

&quot;Lowland Maya civilization flourished from 1000 BCE to 1500 CE in and around the Yucatan Peninsula.&quot;&lt;ref name=Canuto/&gt;

&quot;In 2016, the Pacunam Lidar Initiative (PLI) undertook the largest lidar survey to date of the Maya region, mapping 2144 km&lt;sup&gt;2&lt;/sup&gt; of the Maya Biosphere Reserve in Guatemala.&quot;&lt;ref name=Canuto/&gt;

&quot;Analysis identified 61,480 ancient structures in the survey region, resulting in a density of 29 structures/km&lt;sup&gt;2&lt;/sup&gt;. Controlling for a number of complex variables, we estimate an average density of ~80 to 120 persons/km&lt;sup&gt;2&lt;/sup&gt; at the height of the Late Classic period (650 to 800 CE). Extrapolation of this settlement density to the entire 95,000 km&lt;sup&gt;2&lt;/sup&gt; of the central lowlands produces a population range of 7 million to 11 million.&quot;&lt;ref name=Canuto/&gt;

&quot;Settlement distribution is not homogeneous, however; we found evidence of (i) rural areas with low overall density, (ii) periurban zones with small urban centers and dispersed populations, and (iii) urban zones where a single, large city integrated a wider population.&quot;&lt;ref name=Canuto/&gt;

&quot;The PLI survey revealed a landscape heavily modified for intensive agriculture, necessary to sustain populations on this scale. Lidar shows field systems in the low-lying wetlands and terraces in the upland areas. The scale of wetland systems and their association with dense populations suggest centralized planning, whereas upland terraces cluster around residences, implying local management. Analysis identified 362 km&lt;sup&gt;2&lt;/sup&gt; of deliberately modified agricultural terrain and another 952 km&lt;sup&gt;2&lt;/sup&gt; of unmodified uplands for potential swidden use. Approximately 10&lt;sup&gt;6&lt;/sup&gt; km of causeways within and between sites constitute evidence of inter- and intracommunity connectivity. In contrast, sizable defensive features point to societal disconnection and large-scale conflict.&quot;&lt;ref name=Canuto/&gt;

“The new lidar data show that interconnected Maya cities go back to at least 300 B.C.”&lt;ref name=Chase&gt;{{ cite book
|author=Arlen Chase
|title=Laser mapping shows the surprising complexity of the Maya civilization
|publisher=Science News
|location=
|date=September 27, 2018
|url=https://www.sciencenews.org/article/laser-mapping-shows-surprising-complexity-maya-civilization?utm_source=email&amp;utm_medium=email&amp;utm_campaign=latest-newsletter-v2
|accessdate=7 October 2018 }}&lt;/ref&gt;

The largest structure in the lidar image on the lower right is the Mayan temple at Tikal. The second image down on the right shows the Mayan temple at Tikal from the front.
{{clear}}

==Prehistory==
[[Image:Prehistory at Nairobi National Museum.jpg|right|thumb|300px|Prehistory is diaplayed at Nairobi National Museum. Credit: ninara.{{tlx|free media}}]]
The '''prehistory''' period dates from around 7 x 10&lt;sup&gt;6&lt;/sup&gt; b2k to about 7,000 b2k.

'''Def.''' the &quot;[[wikt:history|history]] of [[wikt:human|human]] [[wikt:culture|culture]] [[wikt:prior|prior]] to [[wikt:written|written]] [[wikt:record|record]]s&quot;&lt;ref name=PrehistoryWikt&gt;{{ cite book
|author=[[wikt:User:SemperBlotto|SemperBlotto]]
|title=prehistory
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=21 December 2014
|url=http://en.wiktionary.org/wiki/prehistory
|accessdate=2015-03-28 }}&lt;/ref&gt; is called '''prehistory'''.
{{clear}}

==21st Century==

&quot;During an underwater expedition in 2000, divers saw a large stone head emerge from the murky dark waters [over Heracleion]. It was the head of the god Hapi, the personification of the Nile’s annual flood.&quot;&lt;ref name=Moss/&gt;

Heracleion, also known by its Egyptian name Thonis,&lt;ref&gt;{{Cite web|url=https://www.trismegistos.org/place/4565|title=TM Places|website=www.trismegistos.org|access-date=2019-05-26}}&lt;/ref&gt; and sometimes called Thonis-Heracleion, was an ancient Egyptian city located near the Canopic Mouth of the Nile, about 32&amp;nbsp;km northeast of Alexandria. Its ruins are located in Abu Qir Bay, currently 2.5&amp;nbsp;km off the coast, under {{convert|10|m|abbr=on|-1}} of water.&lt;ref&gt;{{cite news|url=http://www.huffingtonpost.com/2013/04/29/heracleion-photos-lost-egyptian-city_n_3178208.htm|title=Heracleion Photos: Lost Egyptian City Revealed After 1,200 Years Under Sea|work=The Huffington Post|date=29 April 2013}}&lt;/ref&gt; A stele found on the site indicates that it was one single city known by both its Egyptian and Greek names.&lt;ref name=LostCities6&gt;{{cite web|url=https://www.theguardian.com/cities/2016/aug/15/lost-cities-6-thonis-heracleion-egypt-sunken-sea|title=Lost cities #6: how Thonis-Heracleion resurfaced after 1,000 years under water|last=Shenker|first=Jack|date=15 Aug 2016|newspaper=The Guardian|archive-url=|archive-date=|access-date=11 Feb 2018}}&lt;/ref&gt;

==20th Century==
[[Image:Canopus menouthis herakleion.jpg|right|thumb|300px|Map shows Nile Delta with ancient Canopus, Heracleion, and Menouthis. Credit: [[:de:user:ChrisOsenbrück|ChrisOsenbrück]].{{tlx|free media}}]]
&quot;In 1933 British RAF Group-Captain Cull was flying his plane over Aboukir, a Royal Air Force base east of Alexandria in Egypt, when he glimpsed something in the water below him. From his vantage point, Cull could make out the outlines of structures beneath the water. Unbeknownst to him, Cull had located Heracleion, an important ancient Egyptian city that had lain hidden beneath the water for nearly 1500 years.&quot;&lt;ref name=Moss/&gt;

Heracleion is located at 31°18'15&quot;N 30°06'02&quot;E.

&quot;According to legend, this lost metropolis had hosted its namesake, Heracles, and lovers Paris and Helen before they fled to Troy. Cleopatra, Egypt’s most famous queen, had even been crowned in one of the temples there.&quot;&lt;ref name=Moss/&gt;

&quot;Before its discovery, Heracleion (which was also known in the ancient world by its Egyptian name, Thonis) was almost the stuff of mythology. Though it is now buried several miles off the coast, Thonis-Heracleion was once a thriving port city.&quot;&lt;ref name=Moss/&gt;

The ruins submerged in the sea were located and explored by the French underwater archaeologist Franck Goddio in 1999, after a five-year search.&lt;ref&gt;{{cite web|url=http://www.geeksvip.com/g/21-underwater-discoveries-that-will-make-your-jaw-drop/3/?ipp=1|title=You won't believe what they found under the sea!|website=Geeks VIP|date=12 July 2015|archiveurl=https://web.archive.org/web/20180814232615/http://www.geeksvip.com/g/21-underwater-discoveries-that-will-make-your-jaw-drop/3/?ipp=1|archivedate=14 August 2018}}&lt;/ref&gt;
{{clear}}

==19th Century==
[[Image:Covered Bridge Cedarburg WI May-09.jpg|right|thumb|300px|This Covered Bridge is a latticed timber truss bridge built in 1867 as of May 2009. Credit: [[commons:User:Freekee|Freekee]].{{tlx|free media}}]]
The Covered Bridge on the right was built in 1867 near the town of Cedarburg, Wisconsin.

&quot;In 1866 Mahmoud Bey El-Falaki, the official astronomer to the Viceroy of Egypt, had published a map that located the nearby ancient town of [...] Canopus on the edge of the coastline.&quot;&lt;ref name=Moss/&gt;
{{clear}}

==18th Century==
[[Image:Jantar Mantar at Jaipur.jpg|thumb|right|300px|The Jantar Mantar is a collection of architectural astronomical instruments, built by Maharaja Jai Singh II at his then new capital of Jaipur between 1727 and 1733. Credit: [[w:User:Knowledge Seeker|Knowledge Seeker]].]]
Taken from the observation platform at the top of the Jantar Mantar, the image on the right shows smaller architectural sundials. The Jantar Mantar is a collection of architectural astronomical instruments, built by Maharaja Jai Singh II at his then new capital of Jaipur between 1727 and 1733. The City Palace is behind then Govindji Temple. Nahargarh Fort is on the Hill.
{{clear}}

==17th Century==
[[Image:Kepschem.png|thumb|right|300px|This is a schematic of a Keplerian refracting telescope which uses two different sizes of plano-convex lenses. Credit: Szőcs Tamás.]]
The '''Keplerian Telescope''' was invented by Johannes Kepler in 1611.&lt;ref name=Tunnacliffe&gt;{{ cite book
|title= Optics
|author= AH Tunnacliffe, JG Hirst
|date= 1996
|publisher= 
|location= Kent, England 
|isbn= 0-900099-15-1
|pages= 233–7
|url= }}&lt;/ref&gt;
{{clear}}

==16th Century==
[[Image:Armillary sphere, 1547.jpg|thumb|right|250px|This armillary sphere is dated from 1547. Credit: [[commons:User:Mike Peel|Mike Peel]].]]
[[Image:Angamuco Mexico.jpg|right|thumb|300px|LiDAR gave the power to see underneath the lava that covers Angamuco. Credit: Chris Fisher.{{tlx|fairuse}}]]
The images on the right shows the sphere without sighting tubes or any device for observing astronomical objects and dates from 1547.

Angamuco &quot;occupied 26 square kilometers of land instead of 13 square kilometers.&quot;&lt;ref name=Moon&gt;{{ cite book
|author=Mariella Moon
|title=Ancient city's LiDAR scans reveal as many buildings as Manhattan
|publisher=Yahoo News
|location=
|date=17 February 2018
|url=https://www.yahoo.com/news/ancient-city-apos-lidar-scans-013300196.html?.tsrc=daily_mail&amp;uh_test=2_13
|accessdate=18 February 2018 }}&lt;/ref&gt;

&quot;That is a huge area with a lot of people and a lot of architectural foundations that are represented.&quot;&lt;ref name=Fisher&gt;{{ cite book
|author=Chris Fisher
|title=Ancient city's LiDAR scans reveal as many buildings as Manhattan
|publisher=Yahoo News
|location=
|date=17 February 2018
|url=https://www.yahoo.com/news/ancient-city-apos-lidar-scans-013300196.html?.tsrc=daily_mail&amp;uh_test=2_13
|accessdate=18 February 2018 }}&lt;/ref&gt;

&quot;If you do the maths, all of a sudden you are talking about 40,000 building foundations up there, which is [about] the same number of building foundations that are on the island of Manhattan.&quot;&lt;ref name=Fisher/&gt;

Angamuco &quot;had an unusual layout, with big structures like pyramids and open plazas situated around the edges rather than in the center.&quot;&lt;ref name=Moon/&gt;

&quot;The Purépecha people existed at the same time as the Aztecs. While they are nowhere near as popular as their rivals, they were still a major civilization and had an imperial capital called Tzintzuntzan in western Mexico. Based on [...] LiDAR scans, though, Angamuco is even bigger Tzintzuntzan. It likely wasn't as densely populated, but [...] it's now the biggest city in western Mexico during that period that we know of.&quot;&lt;ref name=Moon/&gt;

&quot;In I523 Cortes quietly appropriated for himself the great Tarascan-held silver district of Tamazula (Jalisco).&quot;&lt;ref name=Sauer&gt;{{ cite journal
|author=Carl O. Sauer
|title=The personality of Mexico
|journal=Geographical Review
|month=July
|year=1941
|volume=31
|issue=3
|pages=353-364
|url=http://www.jstor.org/stable/210171
|arxiv=
|bibcode=
|doi=10.2307/210171
|pmid=
|accessdate=2018-2-18 }}&lt;/ref&gt;
{{clear}}

Vázquez de Coronado set out from Compostela on February 23, 1540, at the head of a much larger expedition composed of about 400 European men-at-arms (mostly Spaniards), 1,300 to 2,000 Mexican Indian allies, four Franciscan friars (the most notable of whom were Juan de Padilla and the newly appointed provincial superior of the Franciscan order in the New World, Marcos de Niza), and several slaves, both natives and Africans.&lt;ref&gt;Winship. pp. 32–4, 37&lt;/ref&gt;&lt;ref&gt;{{Cite journal
|title=What They Never Told You about the Coronado Expedition
|first=R.
|last= Flint
|journal=Kiva
|volume=71
|issue=2
|date=Winter 2005
|pages=203–217
| jstor = 30246725 }}&lt;/ref&gt;
{{clear}}

==15th Century==
[[Image:Sandro Botticelli 052.jpg|thumb|right|250px|An armillary sphere is shown in a painting by Sandro Botticelli, c. 1480. Credit: [http://mail.wikipedia.org/pipermail/wikide-l/2005-April/012195.html Creator: Sandro Botticelli].]]
The image at the right dating from c. 1480 shows the sphere without sighting tube or any device for observing astronomical objects.
{{clear}}

==14th Century==
[[Image:Sceptre de Charles V.jpg|thumb|left|300px|The image shows Charlemagne holding a sceptre dated from before 1380. Credit: [[commons:User:Siren-Com|Siren-Com]].]]
The image at the left shows a sceptre dated to before 1380.

&quot;According to written records, a steady succession of earthquakes, perhaps as many as 23, struck North Africa between A.D. 323-1303.&quot;&lt;ref name=Moss&gt;{{ cite web
|author=Candida Moss
|title=Treasure Trove Discovered at Egypt’s Atlantis, Where Cleopatra Was Crowned
|publisher=Yahoo! News
|location=
|date=4 August 2019
|url=https://news.yahoo.com/treasure-trove-discovered-egypt-atlantis-094651715.html?.tsrc=daily_mail&amp;uh_test=1_04
|accessdate=5 August 2019 }}&lt;/ref&gt;
{{clear}}

==13th Century==
[[Image:Canyons of the Ancients night ruin.jpg|right|thumb|300px|This Anasazi ruin tower is found in the Canyon of the Ancients, New Mexico. Credit: Bob Wick.{{tlx|free media}}]]
The ruin tower on the right is apparently dated to the 13th Century and was built by the Anasazi.
{{clear}}

==12th Century==
[[Image:Iglesia de San Tirso de Sahagún.jpg|right|thumb|300px|Exterior is of the Church of San Tirso, Sahagún.  Credit: [http://flickr.com/photos/17364971@N00 José Antonio Gil Martínez].{{tlx|free media}}]]
Romanesque apses and brick towers of the Church of San Tirso, Sahagún, are shown on the right, dated to the 12th Century.
{{clear}}

==11th Century==
[[Image:Su Song Star Map 1.JPG|thumb|right|300px|This star map is from Su Song's ''Xin Yi Xiang Fa Yao''. Credit: [[commons:User:PericlesofAthens|PericlesofAthens]].]]
The star map on the right, which features a cylindrical projection, was published in 1092 and has a corrected position for the pole star using Shen Kuo's astronomical observations.&lt;ref name=Needham&gt;{{ cite book
|author=Joseph Needham
|date=1986
|title=Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth
|location=Taipei
|publisher=Caves Books Ltd
|pages=208 }}&lt;/ref&gt;
{{clear}}

==10th Century==
[[Image:Visby 11-.JPG|right|thumb|300px|Photo was taken in Visby in August 2007. Credit: [[commons:User:Żeglarz|Żeglarz]].{{tlx|free media}}]]
[[Image:Paektu-san.jpg|right|thumb|300px|Baekdu Mountain—Baitoushan volcano (Paektu-san) is in the Changbai Mountains along the border of today's People's Republic of China and the Democratic People's Republic of Korea in Northeast Asia. Credit: NASA.]]
[[Image:Coins Early Middle Ages.png|right|thumb|300px|Anglo-Saxon rulers wear Roman diadems. Credit: Gunnar Heinsohn.{{tlx|fairuse}}]]
[[Image:Territories of Dynasties in China.gif|right|thumb|300px|The approximate territories of dynasties includes the Jin (China). Credit: Ian Kiu.]]
[[Image:Authentic Viking recreation.jpg|left|thumb|300px|Reconstructions are for the Viking colonisation site at L’Anse aux Meadows National Historic Site, Newfoundland. Credit: Dylan Kereluk.{{tlx|free media}}]]
Visby was founded in the 10th century, on the then independent Baltic Sea island of Gotland. The Hansaetic League formed it during ensuing centuries, during which it came to Denmark. In 1645, it came into Swedish occupation, in which it has remained until today.

There is more about lenses more recently from Visby, Gotland.

&quot;What intrigues the researchers is that the lenses are of such high quality that they could have been used to make a telescope some 500 years before the first known crude telescopes were constructed in Europe in the last few years of the 16th century.&quot;&lt;ref name=Whitehouse&gt;{{ cite book
|author=David Whitehouse
|title=Did the Vikings make a telescope?
|publisher=BBC News
|location=
|date=April 5, 2000
|url=http://news.bbc.co.uk/2/hi/science/nature/702478.stm
|accessdate=2012-10-03 }}&lt;/ref&gt;

&quot;Made from rock-crystal, the lenses have an accurate shape that betrays the work of a master craftsman. The best example of the lenses measures 50 mm (2 inches) in diameter and 30 mm (1 inch) thick at its centre.&quot;&lt;ref name=Whitehouse/&gt;

&quot;The [Visby] Gotland crystals provide the first evidence that sophisticated lens-making techniques were being used by craftsmen over a 1,000 years ago.&quot;&lt;ref name=Whitehouse/&gt;

&quot;Dark and rainy years and harsh winter [occurred] in 913 in Ireland, Switzerland, and France [...] GISP2 shows a 61 ppb peak in SO&lt;sub&gt;4&lt;/sub&gt; sediment in “913–14,” which corresponds to this winter [image on the lowest right in the 8th century section]. Ireland’s increased precipitation and dark could certainly correlate with the presence of volcanic aerosols by virtue of their capacity to act as cloud condensation nuclei.&quot;&lt;ref name=McCormick&gt;{{ cite journal
|author=Michael McCormick, Paul Edward Dutton, and Paul A. Mayewski
|title=Volcanoes and the Climate Forcing of Carolingian Europe, a.d. 750–950
|journal=Speculum A Journal of Medieval Studies
|month=October
|year=2007
|volume=82
|issue=4
|pages=865-895
|url=http://web-static-aws.seas.harvard.edu/climate/seminars/pdfs/mccormick_07.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2017-10-22 }}&lt;/ref&gt;

There &quot;appears to be evidence for a major outbreak of [''Yersinia pestis'']-plague peaking at the end of the “733–960 AD”&lt;sup&gt;4&lt;/sup&gt; time span.&quot;&lt;ref name=HeinsohnTenth&gt;{{ cite journal
|author=Gunnar Heinsohn
|title=TENTH CENTURY COLLAPSE
|journal=Q-Magazine
|month=February
|year=2017
|volume=
|issue=
|pages=1-26
|url=http://www.q-mag.org/_iserv/dlfiles/dl.php?ddl=q-mag-gunnar-10thcentury.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2017-04-01 }}&lt;/ref&gt;

&quot;At Birka, [near Stockholm, Sweden] “a sea level drop estimated up to 5 m has separated the lake from the nearby Baltic Sea of which it was once an inlet, and resulted in the harbour structures being located considerably inland as compared to their original situation”&lt;sup&gt;7&lt;/sup&gt;&quot;&lt;ref name=HeinsohnTenth/&gt; From coin finds, Birka was abandoned around 960.&lt;ref name=Lindqvist&gt;Lindqvist, Herman. ''Historien om Sverige. Islossning till kungarike.'' 1996. See page 165.&lt;/ref&gt;

&quot;Truso [around Hansdorf near Elbing, situated on Lake Drużno near the Baltic Sea just east of the Vistula River] had undergone “isostatic adjustments (vertical crustal movements) [and] eustatic movements (fluctuations in the sea level due to climatic changes). / [The] in-fill consisted of a layer of black/brown sand with a high content of charcoal and ash”&lt;sup&gt;8&lt;/sup&gt;.&quot;&lt;ref name=HeinsohnTenth/&gt;

&quot;In [Great Moravia] some 30 major fortresses, at least nine of them with stone churches, are utterly devastated: &quot;The most recent burnt horizons give evidence for a gigantic annihilation that is roughly datable to the time of 900 CE“&lt;sup&gt;11&lt;/sup&gt;. More recently, the demise of the Great Moravia Empire is dated into the early part of the 10th century&quot;&lt;sup&gt;12&lt;/sup&gt;.&quot;&lt;ref name=HeinsohnTenth/&gt;

&quot;Salzburg, [Austria]’s most important Early Medieval center, becomes “multiple times smaller”&lt;sup&gt;13&lt;/sup&gt; after a devastation in the 10th century when it resorts to primitive wooden houses for the few survivors.&lt;sup&gt;14&lt;/sup&gt;&quot;&lt;ref name=HeinsohnTenth/&gt;

&quot;“There was a rapid, sometimes catastrophic, collapse of many of the pre-existing tribal centers. These events were accompanied by the permanent or temporary depopulation of former areas of settlement. Within a short time new centers representative of the Piast state arose on new sites, thus beginning [in 966] the thousand-year history of the Polish nation and state.”&lt;sup&gt;15&lt;/sup&gt; In the future Piast realm “the local traditional territorial structure was undergoing deep and dramatic changes. Actions which resulted in the abandonment of some of the old strongholds and the building in their place of new ones were associated irrevocably with mass population movement, […] the emergence of new forms and zones of settlement“&lt;sup&gt;16&lt;/sup&gt;. Previously unsettled areas “became densely settled and strongholds appeared; in the second quarter of the tenth century, these were built on a unified model in Bnin, Giecz, Gniezno, Grzybowo, Ostrów Lednicki, Poznan and Smarzewo“&lt;sup&gt;17&lt;/sup&gt;.&quot;&lt;ref name=HeinsohnTenth/&gt;

Archaeology &quot;confirms that [Southern Baltic Ports] mysteriously “undergo discontinuity”&lt;sup&gt;18&lt;/sup&gt; in the 10th c. CE. The indigenous names for some of the deserted ports are not known to this very day.&quot;&lt;ref name=HeinsohnTenth/&gt;

&quot;In [Hungary], the Early Medieval town of Mosaburg with its strikingly Roman style stone Basilica of Zalavár-Récéskút (9th/10th c.) “had become ruinous by the Arpadian age. / Dateable finds from the multilayer cemetery could all be dated to the years from the second third or middle of the 9th century to the early 10th century, namely to its first few decades. / / Not just Mosaburg/Zalavár became depopulated, but also its surrounding area“&lt;sup&gt;19&lt;/sup&gt;.&quot;&lt;ref name=HeinsohnTenth/&gt;

Bulgaria &quot;had the most splendid 9th/10th c. Slavic cities that – to the excavators‘ surprise – had been built in the 700 year earlier style of Rome’s 2nd/3rd c. CE period. Notwithstanding all their stone and brick massiveness, its metropolis, Pliska, comes to a terrible end: “A dark grey (most probably erosion) layer“20 (Henning 2007, 219; bold GH) had strangled that urban jewel for good [...] “Between the 11th and 15th c. CE, [Bulgaria’s; GH] Pliska basin was turned into a desert landscape“&lt;sup&gt;22&lt;/sup&gt;.&quot;&lt;ref name=HeinsohnTenth/&gt;

The Classic Maya &quot;culture of the [Yucatan] collapsed around the same time&lt;sup&gt;25&lt;/sup&gt; (or [Tiwanaku/Bolivia] dated to ca. 1000 CE&lt;sup&gt;26&lt;/sup&gt;)&quot;&lt;ref name=HeinsohnTenth/&gt;

“In Baghdad, the first half of the tenth century had a greater frequency of significant climate events and more intense cold than today, and probably also than the ninth century and the second half of the tenth century”&lt;sup&gt;27&lt;/sup&gt;.&lt;ref name=HeinsohnTenth/&gt;

&quot;The destruction of [Constantinople] must have taken place in the early 10th century when the Port of Theodosius was covered by mud.&quot;&lt;ref name=HeinsohnTenth/&gt;

&quot;Egypt’s most famous export item, writing material made of sheets of papyrus (''Cyperus papyrus'' or Nile grass) ceased to be cultivated around the 10th c. CE&lt;sup&gt;43&lt;/sup&gt;: “All in all, we can say that after the 11th century no writing materials were produced from the papyrus plant&quot;&lt;sup&gt;44&lt;/sup&gt;. The plant had been virtually wiped out&quot;.&lt;ref name=HeinsohnTenth/&gt;

&quot;The collapse of the [Balhae Empire was established under the name Jin] (Chinese: Bohei), stretching from [North Korea via China to Manchuria], is conventionally dated to 926 CE. It should have been noticed in Japan. Yet, a chronicle from a Japanese temple that reports &quot;white ash falling like snow&quot; is currently dated to 946. A recent survey tries to tie the explosion of Changbaishan volcano (also called Mount Paektu) –– located in Southern China close to North Korea, i.e., within the borders of the Balhae Empire –– to the chronicle’s observation:&quot;&lt;ref name=HeinsohnTenth/&gt;

&quot;The written sources document one exceptionally severe winter that affected Europe and western Asia at least as far east as Baghdad in 927–28 but that does not correspond to a major surge in SO&lt;sub&gt;4&lt;/sub&gt; deposits in GISP2.&lt;sup&gt;80&lt;/sup&gt; [The] written sources do not record exceptionally harsh winters for two peaks in GISP2 SO&lt;sub&gt;4&lt;/sub&gt; deposits around “757” and “900–902” [image on the lowest right in the 8th century section].&quot;&lt;ref name=McCormick/&gt;

“The Millennium eruption has fascinated scientists and historians for decades because of its size, potential worldwide impacts. […] Its eruption in 946 was one of the most violent of the last two thousand years and is thought to have discharged around 100 cubic kilometers of ash and pumice into the atmosphere –– enough to bury the entire UK knee deep.&quot;&lt;sup&gt;46&lt;/sup&gt;&lt;ref name=HeinsohnTenth/&gt;

&quot;Eldgjá, has created the largest volcanic canyon in the world. It is some 40 km long, 270 m deep and 600 m wide. The eruption (dated to 934 or 939 CE) resulted in the most massive formation of flood basalt in historical time. 219 million tons of sulfur dioxide were blown into the atmosphere where they reacted with water and oxygen and became 450 million tons of sulfuric acid. These corrosive aerosols must have covered a large part of the Northern Hemisphere&lt;sup&gt;47&lt;/sup&gt;.&quot;&lt;ref name=HeinsohnTenth/&gt;

&quot;A very harsh winter in Germany and Switzerland, and possibly the Low Countries and Ireland, [occurred] in 939–40 [...] The GISP2 layers assigned to “936,” “938,” and “939” show the highest spike in sulfate deposits in the two hundred years under review, 132 ppb [image on the lowest right in the 8th century section]. We consider them to reflect the same event, since they fall within the 2.5 year chronological resolution of individual strata. In this case, tephra have been used to identify chemically the volcano from which the sulfate very probably originated: Eldgjá, on Iceland. The proximity of the Greenland ice sheet to Eldgjá explains the exceptionally high levels of sediment connected with this eruption, for they could well stem from localized tropospheric transport (i.e., lower in the atmosphere) rather than the stratospheric transport, which typically has the most powerful cooling effect.&lt;sup&gt;79&lt;/sup&gt; If it is correct that the ''Chronicon Scotorum'' preserves an eyewitness account of this eruption’s plume in 939, then the connection and the date seem confirmed.&quot;&lt;ref name=McCormick/&gt;

“Throughout the Mediterranean Basin, the Levant, Iran, and southeast Arabia, many valleys display two alluvial fills of which the older dates from about 30,000-10,000 yr BP and the younger from about A.D. 400-1850. […] The younger fill is well sorted and stratified and, as in Mexico, displays silt-clay depletion as well as iron loss when compared with the older fill deposits from which it is often derived. […]. The younger fill is seen in many widely separated areas to cover structures of Roman age as the period of deposition extended into Byzantine and even medieval times. […] The sections in W. Libya are typical in showing the younger fill deposits in channels eroded into the earlier fill. In most areas, the surface of the older fill was the usable land in Roman times. Greek, Roman, Byzantine, and medieval sherds are found in the younger fill, which also covers entire cities, notably, Olympia in Greece.&lt;sup&gt;49&lt;/sup&gt;”.&lt;ref name=HeinsohnTenth/&gt;

&quot;The history of the Anglo-Saxon court is largely lost and unknown.&quot;&lt;ref name=Campbell&gt;{{ cite book
|author=J. Campbell
|title=Anglo-Saxon Courts, In: ''Court Culture in the Early Middle Ages: The Proceedings of the First Alcuin Conference''
|editor=C. Cubitt Hg.
|publisher=Turnhout
|location=Belgium: Brepols
|pages=155-169
|date=2003
|url=http://www.q-mag.org/arthur-of-camelot-and-aththe-of-camulodunum.html
|accessdate=2017-06-21 }}&lt;/ref&gt;

&quot;The Anglo-Saxons, from homelands [in Germany] where the necessary materials scarcely existed, probably had no tradition of building in stone.&quot;&lt;ref name=Heritage&gt;{{ cite book
|author=English Heritage
|title=Story of England. Dark Ages: c 410-1066
|year=2017
|url=http://www.english-heritage.org.uk/learn/story-of-england/dark-ages/architecture/
|accessdate=2017-06-21 }}&lt;/ref&gt;

&quot;Attempts to demonstrate conclusively significant continuity in specific urban or rural sites have run afoul of the near archaeological invisibility of post-Roman British society.&quot;&lt;ref name=Jones&gt;{{ cite book
|author=M.E. Jones
|title=The End of Roman Britain
|publisher=Cornell University Press
|location=Ithaca, NY
|pages=23
|date=1998
|url=http://www.q-mag.org/arthur-of-camelot-and-aththe-of-camulodunum.html
|accessdate=2017-06-21 }}&lt;/ref&gt;

&quot;Whatever the discussion about the plough in Roman Britain, at least it is a discussion based on surviving models and parts of ploughs, whereas virtually no such evidence exists for the Period A.D. 500-900 in England. [...] In contrast to the field system of the 500 years or so on either side of the beginning of our era, little evidence has survived in the ground for the next half millennium.&quot;&lt;ref name=Fowler&gt;{{ cite book
|author=P.J. Fowler
|title=Farming in the First Millennium A.D.: British Agriculture Between Julius Caesar and William the Conqueror
|publisher=Cambridge University Press
|location=Cambridge, UK
|pages=28
|year=2002
|url=http://www.q-mag.org/arthur-of-camelot-and-aththe-of-camulodunum.html
|accessdate=2017-06-21 }}&lt;/ref&gt;

&quot;After all, Alfred the Great (871-899) as well as other Anglo-Saxon rulers take pride in wearing a Roman diadem and/or a Roman chlamys. Offa of Mercia (757-796), e.g., issued a coin that shows him &quot;in the style of a Roman emperor with an imperial diadem in his hair.&quot; [See the coin images third down on the right.]&quot;&lt;ref name=Heinsohn2017&gt;{{ cite book
|author=Gunnar Heinsohn
|title=ARTHUR OF CAMELOT AND ATHTHE-DOMAROS OF CAMULODUNUM: A STRATIGRAPHY-BASED EQUATION PROVIDING A NEW CHRONOLOGY FOR 1st MIILLENNIUM ENGLAND
|publisher=Quantavolution Magazine
|location=
|date=15 June 2017
|url=http://www.q-mag.org/arthur-of-camelot-and-aththe-of-camulodunum.html
|accessdate=2017-06-21 }}&lt;/ref&gt;

Embossed &quot;clay vessels attributed to Angle-Saxons follow the pattern of Anglo-Saxon coinage because they, too, point to a “deliberate imitation of Roman silver or glass ware” of the 1st/2nd century (Myres 1969, 30).&quot;&lt;ref name=Heinsohn2017/&gt;

There are &quot;the rich Roman strata in Anglo-Saxon capitals, like Alfred’s (''Venta Belgarum'') (Winchester)&quot;.&lt;ref name=Heinsohn2017/&gt;

For &quot;authors of the 9th century AD, like Harun ibn Yahya, a Syrian traveler writing in 866, there is no doubt that Britain (''Bartīniyah'') is the “the last of the lands of the Greeks [Rum/Romans], and there is no civilization beyond them” (Green 2016).&quot;&lt;ref name=Heinsohn2017/&gt;

The unknown author of &quot;the Persian ''Hudud al-'Alam'' (982 AD) in which Britain (al-Baritiniya) &quot;is the last land of Rum [Rome] on the coast of the Ocean&quot; (Watson 2001).&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Stratigraphically, there is no problem with such a statement since—between the year 1 and the 930s AD—there are only enough building strata with streets, residential quarters, latrines, aqueducts etc. for a period of some 230 Roman years in Britain. Since they are contingent with the High Middle Ages of the 10th century AD, these massive Roman strata cannot help but belong to the 8th-10th century period, whatever the textbook chronology requires.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Everything we know from Early Medieval texts pertaining to 8th/9th century Anglo-Saxons confirms that they thrived in a classical culture, in a genuine Roman environment. That makes sense only when the hard evidence of the period dated 1st to 3rd century receives the 8th-10th century dates of its stratigraphic location immediately before the onset of the High Middle in the 10th century AD:&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Anglo-Saxon England was peopled with learned men and women, highly educated in Latin and English, who circulated and read Classical texts as well as composing their own. [...] There survives a large corpus of literature showing a deep understanding of the physical and the metaphysical [...]. Charters show that laws, administration and learning were not just for an educated elite. Laypeople were involved in the ceremonies and had documents created for them: land grants, wills, dispute settlements. [...] The coinage across the period shows an elaborate and controlled economy. This was a well-managed society not given to lawlessness and chaos. [...] They drew influence from Classical art and developed their own distinct artistic styles. [...] They had trade routes stretching across the known world and were familiar with and able to buy spices, pigments and cloth from thousands of miles away (many manuscripts use a blue pigment made from lapis lazuli, brought from Afghanistan. [...] The English church was in close contact with Rome, with correspondence travelling back and forth; new bishops would be sent to Rome to collect the pallium; and King Alfred visited the city as a young boy.&quot;&lt;ref name=Wiles&gt;{{ cite journal
|author=K. Wiles
|title=Back to the Dark Ages
|journal=History Today
|issue=
|volume=
|pages=
|month=5 May
|year=2016
|url=http://www.historytoday.com/kate-wiles/back-dark-ages
|accessdate=2017-06-21 }}&lt;/ref&gt;

The Norse settlement of Vinland at L’Anse aux Meadows National Historic Site, Newfoundland, in the image on the left, has been radiocarbon dated to c. 1000, or 1,000 b2k.
{{clear}}

==9th Century==
[[Image:Cairo Nilometer 1.jpg|right|thumb|300px|Nilometer is at Rawda, Cairo, Egypt. Credit: [[commons:User:Baldiri|Baldiri]].{{tlx|free media}}]]
[[Image:Climbing pyram.jpg|left|thumb|300px|People are climbing the stairs of the pyramid on 9 December 2010. Credit: [[commons:User:Leon petrosyan|Leon petrosyan]].{{tlx|free media}}]]
[[Image:Rökstenen 1.JPG|right|thumb|300px|Rök Runestone is located near Ödeshög in south Sweden. Credit: Bengt Olof ÅRADSSON.{{tlx|free media}}]]
[[Image:Rökstenen2.JPG|220px|left|thumb|The front of the stone, the beginning of the inscription is read by tilting your head to the left. Credit: [[w:user:Wiglaf|Wiglaf]].{{tlx|free media}}]]
There was an inscription which placed the foundation of the nilometer in 861.

Cobá is a former pre-Columbian Mayan city on the Yucatán Peninsula southeast of Valladolid located in the state of Quintana Roo, Mexico.

In Cobá, the temple pyramid Nohoch-Mul (also known as Castillo, or the climbing pyramid shown on the left) is 42 meters high.

The city was founded shortly after the beginning of the year and expanded into a city state that peaked between 600 and 800 (1400 and 1200 b2k).

The Rök runestone is considered the first piece of written Swedish literature and thus it marks the beginning of the history of Swedish literature.&lt;ref name=Gustafson1&gt;Gustafson, Alrik, Svenska litteraturens historia, 2 volums (Stockholm, 1963). First published as A History of Swedish Literature (American-Scandinavian Foundation, 1961). Chapter 1.&lt;/ref&gt;&lt;ref&gt;''Forntid och medeltid'', Lönnroth, in Lönnroth, Göransson, Delblanc, ''Den svenska litteraturen'', vol 1.&lt;/ref&gt;

It was proposed that the inscription has nothing at all to do with the recording of heroic sagas and that it contains riddles which refer only to the making of the stone itself.&lt;ref&gt;[http://www.gu.se/english/about_the_university/news-calendar/News_detail//new-interpretation-of-the-rok-runestone-inscription-changes-view-of-viking-age.cid1371180 New interpretation of the Rök runestone inscription changes view of Viking Age] (University of Gothenburg)&lt;/ref&gt;&lt;ref&gt;Holmberg, Per (2015) [http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A922588&amp;dswid=5655 Svaren på Rökstenens gåtor: En socialsemiotisk analys av meningsskapande och rumslighet (&quot;The answers to Rök stone riddles: A social analysis of the meaning and spatiality&quot;)]  Futhark: International Journal of Runic Studies, Vol. 6. pp. 65-106. (in Swedish)&lt;/ref&gt;

The Rök runestone inscription is not connected to heroic deeds in war. Instead it deals with the conflict between light and darkness, warmth and cold, life and death.&lt;ref name=Holmberg&gt;{{ cite journal
|author=Per Holmberg, Bo Gräslund, Olof Sundqvist and Henrik Williams
|title=The Rök Runestone and the End of the World
|journal=Futhark: International Journal of Runic Studies
|date=15 January 2020
|volume=9-10
|issue=
|pages=7-38
|url=https://doi.org/10.33063/diva-401040
|arxiv=
|bibcode=
|doi=10.33063/diva-401040
|pmid=
|accessdate=19 January 2020 }}&lt;/ref&gt;

&quot;The Rök runestone from central middle Sweden, dated to around 800 CE, [...] Combining perspectives and findings from semiotics, philology, archaeology, and history of religion, the study presents a completely new interpretation which follows a unified theme, showing how the monument can be understood in the socio-cultural and religious context of early Viking Age Scandinavia. The inscription consists, according to the pro­posed interpretation, of nine enigmatic questions. Five of the questions con­cern the sun, and four of them, it is argued, ask about issues related to the god Odin. A central finding is that there are relevant parallels to the inscription in early Scandinavian poetry, especially in the Eddic poem Vafþrúðnismál.&quot;&lt;ref name=Holmberg/&gt;
{{clear}}

==8th Century==
[[Image:Dunhuang Star Atlas - complete.jpg|thumb|center|400px|The Dunhuang Star Atlas is the last section of Or.8210/S.3326. Credit: Unknown.]]
[[Image:Dunhuang star map.jpg|thumb|250px|This is an image of the [[w:Dunhuang map|Dunhuang map]] from the [[w:Tang Dynasty|Tang Dynasty]] of the North Polar region. Constellations of the three schools are distinguished with different colors: white, black and yellow for stars of [[w:Wu Xian (astronomer)|Wu Xian]], [[w:Gan De|Gan De]] and [[w:Shi Shen|Shi Shen]] respectively. Credit: [[commons:User:Laurascudder|Laurascudder]], from: Brian J. Ford (1993). Images of Science: A History of Scientific Illustration, Oxford University Press. {{ISBN|0195209834}}.]]
The [[w:Dunhuang map|Dunhuang map]] from the [[w:Tang Dynasty|Tang Dynasty]] of the North Polar region at right is thought to date from the reign of [[w:Emperor Zhongzong of Tang|Emperor Zhongzong of Tang]] (705–710). Constellations of the three schools are distinguished with different colors: white, black and yellow for stars of [[w:Wu Xian (astronomer)|Wu Xian]], [[w:Gan De|Gan De]] and [[w:Shi Shen|Shi Shen]] respectively. The whole set of star maps contains 1,300 stars.

The Dunhuang Star Atlas, the last section of manuscript Or.8210/S.3326. It is &quot;the oldest manuscript star atlas known today from any civilisation, probably dating from around AD 700. It shows a complete representation of the Chinese sky in 13 charts with over 1300 stars named and accurately presented.&quot;&lt;ref name=BritishLibrary&gt;{{ cite book
|author=British Library
|title=The Dunhuang Star Atlas
|publisher=International Dunhuang Project (IDP)
|location=British Library
|date=June 2015
|url=http://idp.bl.uk/4DCGI/education/astronomy/atlas.html
|accessdate=2015-12-27 }}&lt;/ref&gt;

&quot;The Dunhuang Star Atlas [above center] forms the second part of a longer scroll (Or.8210/S.3326) that measures 210 cm long by 24.4 cm wide and is made of fine paper in thirteen separate panels.&quot;&lt;ref name=BritishLibrary/&gt;

&quot;The first part of the scroll is a manual for divination based on the shape of clouds. The twelve charts showing different sections of the sky follow these. The stars are named and there is also explanatory text. The final chart is of the north-polar region. The chart is detailed, showing a total of 1345 stars in 257 clearly marked and named asterisms, or constellations, including all twenty-eight mansions.&quot;&lt;ref name=BritishLibrary/&gt;

&quot;The importance of the chart lies in both its accuracy and graphic quality. The chart includes both bright and faint stars, visible to the naked eye from north central China&quot;.&lt;ref name=BritishLibrary/&gt;
{{clear}}

==7th Century==
[[Image:Sambor Prei Kuk N16 A.jpg|right|thumb|300px|N16 Sanctuary is in Sambor Prei Kuk, Cambodia. Credit: [[commons:User:Baldiri|Baldiri]].{{tlx|free media}}]]
[[Image:Cantona (20065785913).jpg|left|thumb|300px|Cantona is a Mesoamerican archaeological site in the state of Puebla, Mexico. Credit: [https://www.flickr.com/people/67769030@N07 Arian Zwegers from Brussels, Belgium].{{tlx|free media}}]]
Sambor Pre Kuk, with its N16 Sanctuary imaged on the right, is an archaeological complex formed by the remains of the city of Isanapura, the capital of the kingdom of Chenla, an immediate predecessor of the Khmer Empire (pre-Angkorian).

This city was built during the reign of Isanavarman I (616-635). At this time, several constructions, clear predecessors of Khmer architecture, were erected in Angkor.

Cantona is a Mesoamerican archaeological site in the state of Puebla, Mexico. It was a fortified city with a high urbanization level at prehispanic times, probably founded by Olmec-Xicalanca groups towards the late Classical Period. It sat astride an old trading route between the Gulf Coast and the Central Highlands and was a prominent, if isolated, Mesoamerican city between 600 and 1000 CE. After Chichimec's invasions in the 11th century, Cantona was abandoned.

Cantona's inhabitants were mainly agricultural farmers and traders, particularly for obsidian, obtained from Oyameles-Zaragoza mountains surrounding the city. Additionally, they may have been supplying the lowlands with a derivative of the maguey plant, pulque. Cantona's population is estimated at about 80,000 inhabitants at its peak.

Cantona may well be the largest prehispanic city yet discovered in Mesoamerica. Limited archaeological work has been done at the site, and only about 10% of the site can be seen. The Pre-Columbian settlement area occupies approximately 12 km², distributed in three units, of which the largest is at the south, with a surface of 5 km². The site comprises a road network with over 500 cobblestone causeways, more than 3,000 individual patios, residences, and 24 ball courts - more than in any other mesoamerican site. It has an elevated Acropolis over the rest of the city in which the main buildings of the city were built. This was used for the ruling elite and priests, and was where the temples of the most important deities where located. These impressive buildings were constructed with carved stones (one atop the other) without any stucco or cement mortar. Cantona certainly was built with a definite urban design and walkways connecting each and every part of the city. The &quot;First Avenue&quot; is 563 meters in length.
{{clear}}

==6th Century==
[[Image:Basilica Cistern in Istanbul, Turkey (6th century, former Constantinople) (32691506623).jpg|right|thumb|300px|Basilica Cistern is in Constantinople, Turkey, dates from the 6th century. Credit: [https://www.flickr.com/people/145472109@N04 Ralf Steinberger from Northern Italy and Berlin].{{tlx|free media}}]]
[[Image:Dzibanche Buhos2.jpg|thumb|left|300px|The Temple of the Owl pyramid is at Dzibanche. Credit: [[commons:User:HJPD|HJPD]].]]
On the right is the Basilica Cistern in Constantinople, Turkey. It has been dated to the 6th century.

Dzibanche is an archaeological site which includes the Temple of the Owl pyramid. It is an ancient Maya site located in southern Quintana Roo, in the Yucatan Peninsula of southeastern Mexico.

Structures at Dzibanche include the Temple of the Captives, the Temple of the Lintels and the Temple of the Owl, on the left.
{{clear}}

==5th Century==
[[Image:Iron Pillar at Qutub Minar Complex India-0346 - Flickr - archer10 (Dennis).jpg|right|thumb|250px|Iron Pillar is seven-metres high in the courtyard of the mosque and has been there long before the mosque's construction. Credit: [http://www.flickr.com/people/22490717@N02 Dennis Jarvis from Halifax, Canada].{{tlx|free media}}]]
Ancient India was an early leader in metallurgy, as evidenced by the wrought-iron Pillar of Delhi in the image on the right, dated to about 415 or 1585 b2k.
{{clear}}

==4th Century==
[[Image:Capernaum, Israel 06.jpg|right|thumb|300px|This shows the House of Peter, Capernaum, Israel. Credit: [[commons:User:Hoshvilim|Hoshvilim]].{{tlx|free media}}]]
The House of Peter in Capernaum, Israel, has been dated to the 4th century.

&quot;The most severe [earthquake of North Africa] occurred in A.D. 365. The coastline fell and the cluster of cities that lay in the Canopic branch of the Nile vanished into the Mediterranean.&quot;&lt;ref name=Moss/&gt;
{{clear}}

==Imperial Antiquity==
[[Image:Pile from The Strood.png|right|thumb|300px|Pile from The Strood, in Roman cut (223 cm high), re-dated from the late 1st c. AD to the 7th/8th c. AD. Roman lead covered box with Roman glass urn (100-120 CE) from Mersea’s Roman barrow. Credit: Gunnar Heinsohn.{{tlx|fairuse}}]]
&quot;The Strood causeway to Mersea Island was thought to be Roman, built in the 1st c. AD. It leads to Mersea’s Roman burial mound (barrow) where a typical Roman lead covered box with a no less typical Roman glass urn (tentatively dated between 100 and 120 AD) was retrieved [in the image on the right]. Oak piles in typical Roman cut were discovered in 1978. Up to the 1980s it was never doubted that the dam was built by Romans in the 1st c. AD to reach their settlements on the Island.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Scientific dating methods have been applied to some substantial oak piles discovered beneath the Strood in 1978, when a water-main was being laid. They indicate that the structure was probably built between A.D. 684 and 702. The piles were discovered at the south end of the causeway where the trench was at its deepest—they were about 1.6m below the present ground level and were sealed by a series of road surfaces. Seven piles were recovered and samples were submitted to Harwell laboratory for radiocarbon dating to get a rough idea of the date. Samples from four of the piles were sent to the University of Sheffield for tree ring dating (dendrochronology). The remaining three piles are now in the Colchester and Essex Museum. The dating of the construction to AD 684 to 702 was regarded as conclusive.&quot;&lt;ref name=Millat&gt;{{ cite book
|author=T. Millat
|title=Essex Archaeology and History
|publisher=Mersea Museum
|location=Mersea, UK
|year=1982
|url=http://www.merseamuseum.org.uk/mmresdetails.php?tot=1&amp;cat=&amp;col=MM&amp;typ=ID&amp;rt=Article&amp;syn=all&amp;ord=dtadd&amp;hit=0&amp;pid=EAH_014_077
|accessdate=2017-06-21 }}&lt;/ref&gt;

In ''Felix Romuliana'', &quot;the construction [...] is [...] Imperial Antique (1st-3rd c. [1900-1700 b2k]), and sometimes even late Hellenistic, [in] appearance.&quot;&lt;ref name=Heinsohn&gt;{{ cite journal
|author=Gunnar Heinsohn
|title=Felix Romuliana
|journal=Q Magazine
|location=
|month=15 March
|year=2017
|url=http://www.q-mag.org/
|accessdate=2017-05-13 }}&lt;/ref&gt;
{{clear}}

==3rd Century==
[[Image:Dermainzer Globus.jpg|thumb|right|250px|The Mainz celestial globe is the last known celestial globe of Roman antiquity (1850-1780 b2k, 11 cm diameter). Credit: Gunnar Heinsohn.]]
[[Image:Der mainzer Globus.jpg|thumb|left|250px|These are constellation illustrations on the last known celestial globe of Roman antiquity. Credit: Gunnar Heinsohn.]]
The last known celestial globe shown at the right dates from 1850 to 1780 b2k. The constellation illustrations from the Mainz celestial globe are shown at the left.
{{clear}}

==2nd Century==
[[Image:P. Oxy. I 29.jpg|right|thumb|300px|Oxyrhynchus papyrus (P.Oxy. I 29) shows a fragment of Euclid's ''Elements''. Credit: Euclid.{{tlx|free media}}]]
On the right is an image of the oldest extant diagram of Euclid's ''Elements'', found at Oxyrhynchus and dated to c. 100 AD.&lt;ref name=Casselman&gt;{{ cite book
|url=http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html
|title=One of the Oldest Extant Diagrams from Euclid
|author=Bill Casselman
|publisher=University of British Columbia
|accessdate=26 September 2008 }}&lt;/ref&gt;
{{clear}}

==1st Century==

Pompeii lens (plano-convex) from the excavations of the Via Stabia, &quot;House of the Engraver&quot; dated to 79 AD per E. Gerspach, ''L'art de la verrerie'' (Paris 1885) 41-42.

Roman London lens (fragmentary biconvex glass lens of light green color), from Roman London, dated to 43-50 AD per H. Syer Cuming, &quot;On Spectacles,&quot; The Journal of the British Archaeological Association 11 (1855) 144-49.

==-1st Century==
[[Image:MauesBuddhaCoin.JPG|thumb|300px|right|Indian-standard coin of King Maues has on the obverse a rejoicing elephant holding a wreath, symbol of victory. Credit: [[w:user:Per Honor et Gloria|Per Honor et Gloria]]].]]
The &quot;Late La Tène time span [is] between the conquests of 55 BC and 54 BC [2055 and 2054 b2k] under Julius Caesar (100-44 BC) and the time of Christ. In the rare cases where pottery and tableware are attributed to Saxons of the 4th/5th c. AD, &quot;astonishingly La Tène art styles [more than 300 years out of fashion] re-emerge as dominant in the northern and western zone.&quot; (Hines 1996, 260)&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Stamped pottery has had a long and varied history in Britain. There have been periods when it flourished and periods when
it almost totally disappeared. This article considers two variations of the rosette motif (A 5) and their fortunes from the late Iron Age to the Early Saxon period. [...] The La Tène ring stamps [which end in the 1st century BC; GH ] are found in a
range of designs, from the simple negative ring (= AASPS Classification A 1bi) to four concentric negative rings (= AASPS A
2di). These motifs are also found in the early Roman period [1st century AD; GH]. [...] The 'dot rosettes' (= AASPS A 9di)
on bowls from the [Late Latène] Hunsbury hill-fort (Fell 1937) use the same sort of technique as the dimple decoration on
4th-century 'Romano-Saxon' wares.&quot;&lt;ref name=Briscoe&gt;{{ cite journal
|author=D.C. Briscoe
|title=Two Important Stamp Motifs in Roman Britain and Thereafter, In: ''Romano-British Pottery in the Fifth Century''
|journal=Internet Archaeology
|issue=41
|date= 2016
|doi=https://doi.org/10.11141/ia.41.2
|accessdate=2014-10-25 }}&lt;/ref&gt;

In &quot;Šarnjaka kod Šemovca (Dalmatia/Croatia), e.g., contain 700-year-older La Tène and Imperial period items (1st century BC to 3rd
century AD) [...]:&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;A large dugout house (SU 9) was discovered in the course of the investigation in 2006. Its dimensions are 4.8 by 2.1 metres, with a depth of 34 centimetres, and an east-west orientation, deviating slightly along the NE-SW line. It contained numerous sherds of Early Medieval pottery, two fragments of glass, and a small iron spike. Three sherds of Roman pottery [1st-3rd c. CE; GH] and ten sherds of La Tène pottery [ending 1st c. BCE; GH] were also recovered from the house.&quot;&lt;ref name=Bekic&gt;{{ cite journal
|author=L. Bekić
|title=Nalazi 8. i 9. stoljeća sa Šarnjaka kod Šemovca / Finds from the 8th and 9th centuries at Šarnjak near Šemovec
|journal=Vjesnik Arheološkog muzeja u Zagrebu (VAMZ)
|volume=XLIX
|pages=219-248
|date= 2016
|url=
|accessdate=2014-10-25 }}&lt;/ref&gt;

&quot;The contemporaneity of Rome’s Imperial period textbook-dated to the 1st-3rd century AD with the Early Middle Ages (8th-10th century AD) is also confirmed for Poland [in the stratigraphic table above]. There, too, Late Latène (conventionally ending 1st c. BC) immediately precedes the Early Medieval period of the 8th-10th c. CE.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;In [the Roman Empire] capital cities, Rome and Constantinople (Heinsohn 2016) [they] build residential quarters, streets, latrines, aqueducts, ports etc. only in one of the three periods—Imperial Antiquity, Late Antiquity, and Early Middle Ages—dated between 1 and 930s AD. In Rome, they are assigned to Imperial Antiquity (1st-3rd c.); in Constantinople, to Late Antiquity (4th-6th c.).&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Roman churches of Late Antiquity and the Early Middle Ages [...] would suffice to confirm the existence of these two periods. The churches are there. However, we never find churches of the 8th or 9th century superimposed on churches of the 4th or 5th century that, in turn, are superimposed on pagan basilicas of the 1st or 2nd century. They all share the same stratigraphic level of the 1st and 2nd/early 3rd century. Moreover, the ground plans of the 4th/5th—as well as the 8th/9th—century churches slavishly repeat the ground plans of 1st/2nd century basilicas, as already pointed out 75 years ago by Richard Krautheimer (1897-1994). It is this period of Imperial Antiquity (with its internal evolution from the 1st to 3rd centuries) that alone builds the residential quarters, latrines, streets, and aqueducts so desperately looked for in Late Antiquity and the Early Middle Ages. Thus, Rome does not have more stratigraphy for the first millennium AD than England or Poland.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Germanic tribes, not only Anglo-Saxons and Frisians but also Franks, had been competing with Rome for the conquest of the British Isles since the 1st century BC&quot;.&lt;ref name=Heinsohn2017/&gt;

&quot;1st century BC &quot;Astonishingly LA TÈNE art styles&quot; (Hines 1996) dominate pottery of SAXONS [and] Powerful LA TÈNE Celts with King Aththe-Domarous of Camulodunum [is the] greatest ruler.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Saxons begin their attack on Britain as early as the 1st century BC. They compete with the Romans, who may have employed Germanic Franks as auxiliary forces. The Saxons invade from the East, i.e., from the German Bight.&quot;&lt;ref name=Heinsohn2017/&gt;

From &quot;the stratigraphy of the Saxon homeland, located around Bremen/Weser inside today’s Lower Saxony [it] is mainly inhabited by Chauci and Bructeri [...] Saxon tribes that are [...] at war with the Romans in the time of Augustus (31 BC-14 AD) and Aththe-Domaros of Camulodunum (Aθθe-Domaros, also read as Addedom-Arus; c. 15-5 BC).&quot;&lt;ref name=Heinsohn2017/&gt;

On the right is an Indian-standard coin of King Maues. On the obverse is a rejoicing elephant holding a wreath, a symbol of victory. The Greek legend reads ΒΑΣΙΛΕΩΣ&amp;#32;ΒΑΣΙΛΕΩΝ&amp;#32;ΜΕΓΑΛΟΥ&amp;#32;ΜΑΥΟΥ (Great King of Kings Maues). The reverse shows the seated king Maues. Kharoshthi legend: RAJATIRAJASA MAHATASA MOASA (Great King of Kings Maues).
{{clear}}

==-2nd Century==
[[Image:NAMA Machine d'Anticythère 1.jpg|right|thumb|300px|The image shows the main Antikythera mechanism fragment (fragment A). Credit: [[commons:User:Marsyas|Marsyas]].{{tlx|free media}}]]
[[Image:Nike of Samothrake Louvre Ma2369 n4.jpg|right|thumb|250px|The Nike of Samothrace, ca 190 BC, is a masterpiece of Hellenistic art. Credit: Archaeological expedition of Charles Champoiseau, 1863 and 1879.]]
[[Image:Lagid queen Isis Ma3546.jpg|left|thumb|250px|Cleopatra II was born c. 185 BC and died 116 BC. Credit: [[commons:User:Jastrow|Jastrow]].{{tlx|free media}}]]
The &quot;fragments [of the Antikythera Mechanism] contain at least 30 interlocking gear-wheels, along with copious astronomical inscriptions. Before its sojourn on the sea bed, it computed and displayed the movement of the Sun, the Moon and possibly the planets around Earth, and predicted the dates of future eclipses.&quot;&lt;ref name=Marchant&gt;{{ cite journal
|author=Jo Marchant
|title=In search of lost time
|journal=Nature
|date=30 November 2006
|volume=444
|issue=7119
|pages=534-538
|url=https://www.nature.com/nature/journal/v444/n7119/full/444534a.html
|arxiv=
|bibcode=
|doi=10.1038/444534a
|pmid=
|accessdate=2017-11-05 }}&lt;/ref&gt;

The Winged Nike of Samothrace is made from Parian marble, ca. 190 BC? and found in Samothrace in 1863 by the archaeological expedition of Charles Champoiseau, 1863 and 1879.

Cleopatra II on the left was involved in the ruling of Egypt apparently from c. 175 BC to until she died in 116 BC.
{{clear}}

==Iron Age==
[[Image:Three iron beads from Gerzeh.jpg|right|thumb|250px|Photographs of three of the originally nine iron beads from Gerzeh, Lower Egypt, from left UC10738, UC10739 and UC10740. Credit: Thilo Rehren, Tamás Belgya, Albert Jambon, György Káli, Zsolt Kasztovszky, Zoltán Kis, Imre Kovács, Boglárka Maróti, Marcos Martinón-Torres, Gianluca Miniaci, Vincent C. Pigott, Miljana Radivojević, László Rosta, László Szentmiklósi, Zoltán Szőkefalvi-Nagy.]]
[[Image:Neutron radiography.jpg|left|thumb|250px|Comparison of neutron radiography and an optical photograph of an iron bead is shown. Credit: Thilo Rehren, Tamás Belgya, Albert Jambon, György Káli, Zsolt Kasztovszky, Zoltán Kis, Imre Kovács, Boglárka Maróti, Marcos Martinón-Torres, Gianluca Miniaci, Vincent C. Pigott, Miljana Radivojević, László Rosta, László Szentmiklósi, Zoltán Szőkefalvi-Nagy.]]
[[Image:Slag blocks at Otobo Dunoka, Lejja.png|thumb|right|250px|The slag blocks shown are at Otobo Dunoka, Lejja, Nigeria. Credit: Pamela Eze–Uzomaka.{{tlx|fairuse}}]]
The Iron Age in Africa began approximately 2000 BC (4000 b2k) and ended 500 AD (1500 b2k).

The Bantu expansion spread the technology to Eastern and Southern Africa during {{circa}} 500 BCE to 400 CE, as shown in the Urewe culture&lt;ref name=Shillington&gt;K. Shillington, ''History of Africa'' (3rd ed.  2005), [https://books.google.com/books?id=YdscBQAAQBAJ&amp;lpg=PR1&amp;pg=PA61 61&amp;ndash;68].&lt;/ref&gt; of the Lake Victoria region.

The Nok culture of Nigeria may have practiced iron smelting from as early as 1000 BCE.&lt;ref name=Miller1994&gt;Duncan E. Miller and N.J. Van Der Merwe, 'Early Metal Working in Sub Saharan Africa' ''Journal of African History'' 35 (1994) 1-36&lt;/ref&gt;&lt;ref name=Stuiver&gt;Minze Stuiver and N.J. Van Der Merwe, 'Radiocarbon Chronology of the Iron Age in Sub-Saharan Africa' ''Current Anthropology'' 1968. Tylecote 1975&lt;/ref&gt;

&quot;After a typological analysis and a cross-dating of bronze artifacts recovered north and south of the Alps, the Roman school of Peroni set the 1020 [3020 b2k] as the beginning of the Iron Age (De Marinis 2005, p. 21; Pacciarelli 2005). The date is in agreement with the chronology supported by Lothar Sperber (Sperber 1987). The recent works of Nijboer based on the analysis of radiocarbon dates from Latial contexts agree with this high chronology (Nijboer et al. 1999-2000; Nijboer &amp; Van der Plicht 2008; Van der Plicht et al. 2009).&quot;&lt;ref name=Capuzzo/&gt;

Evidence exists for earlier iron metallurgy in parts of Nigeria, Cameroon, and Central Africa, possibly from around 2,000 BCE.&lt;ref name=Holl-2020&gt;{{cite journal|last1=Holl|first1=Augustin F. C.|title=The Origins of African Metallurgies
|journal=Oxford Research Encyclopedias|date=June 2020|volume=22|issue=4|pages=415–438|doi=10.1093/acrefore/9780190854584.013.63|{{isbn|9780190854584}} }}&lt;/ref&gt;

Evidence for Iron smelting in West Africa was found in Lejja, dating back to 2000 BC.&lt;ref name=Uzomaka&gt;{{Cite book|last=Eze–Uzomaka|first=Pamela|title=Iron and its influence on the prehistoric site of Lejja, In: ''The world of iron''
|publisher=Archetype
|location=London
|date=2013
|editor=J. Humphris &amp; T. Rehren
|pages=3-9
|url=https://www.academia.edu/4103707/Iron_and_its_influence_on_the_prehistoric_site_of_Lejja }}&lt;/ref&gt;

The inception of iron metallurgy in Africa occurred between 3000 and 2500 BCE.&lt;ref name=Eggert&gt;{{Cite book|title= Nok: African Sculpture in Archaeological Context|last= Eggert|first= Manfred|publisher= Africa Magna Verlag Press|year= 2014|editor-last= Breunig|editor-first= P|location= Frankfurt, Germany|pages= 1–2|chapter= Early iron in West and Central Africa}}&lt;/ref&gt;&lt;ref name=Holl2020&gt;{{cite journal|last1=Holl|first1=Augustin F. C.|title=The Origins of African Metallurgies
|journal=Oxford Research Encyclopedias|date=June 2020
|volume=22|issue=4|pages=415–438|doi=10.1093/acrefore/9780190854584.013.63|isbn=9780190854584 }}&lt;/ref&gt;

&quot;The earliest known iron artefacts are nine small beads securely dated to circa 3200 BC, from two burials in Gerzeh, northern Egypt.&quot;&lt;ref name=Rehren&gt;{{ cite journal
|author=Thilo Rehrena
|author2=Tamás Belgya
|author3=Albert Jambon
|author4=György Káli
|author5=Zsolt Kasztovszky
|author6=Zoltán Kis
|author7=Imre Kovács
|author8=Boglárka Maróti
|author9=Marcos Martinón-Torres
|author10=Gianluca Miniaci
|author11=Vincent C. Pigott
|author12=Miljana Radivojević
|author13=László Rosta
|author14=László Szentmiklósi
|author15=Zoltán Szőkefalvi-Nagy
|title=5,000 years old Egyptian iron beads made from hammered meteoritic iron
|journal=Journal of Archaeological Science
|month=December
|year=2013
|volume=40
|issue=12
|pages=4785–92
|url=http://www.sciencedirect.com/science/article/pii/S0305440313002057
|arxiv=
|bibcode=
|doi=10.1016/j.jas.2013.06.002
|pmid=
|accessdate=2016-10-23 }}&lt;/ref&gt;

&quot;Since both tombs are securely dated to Naqada IIC–IIIA, c 3400–3100 BC (Adams, 1990: 25; Stevenson, 2009: 11–31), the beads predate the emergence of iron smelting by nearly 2000 years, and other known meteoritic iron artefacts by 500 years or more (Yalçın 1999), giving them an exceptional position in the history of metal use.&quot;&lt;ref name=Rehren/&gt;

The image on the left uses neutron radiography to show the metal underneath the corrosion.

&quot;Bead UC10738 [in the image on the right] has a maximum length of 1.5 cm and a maximum diameter of 1.3 cm, bead UC10739 is 1.7 cm by 0.7 cm, and bead UC10740 is 1.7 cm by 0.3 cm. All three beads are of rust-brown colour with a rough surface, indicative of heavy iron corrosion. Initial analysis by [proton–induced X–ray fluorescence] pXRF indicated an elevated nickel content of the surface of the beads, in the order of a few per cent, and their magnetic property suggested that iron metal may be present in their body (Jambon, 2010).&quot;&lt;ref name=Rehren/&gt;

Meteoric iron, a characteristic iron–nickel alloy, was used by various ancient peoples thousands of years before the Iron Age. Such iron, being in its native metallic state, required no smelting of ores.&lt;ref&gt;[https://books.google.com/books?id=7VMVguiMmY0C&amp;pg=PA164 ''Archaeomineralogy'', p. 164], George Robert Rapp, Springer, 2002&lt;/ref&gt;&lt;ref&gt;[https://books.google.com/books?id=DaAmwiJ4rnEC&amp;pg=PA125 ''Understanding materials science'', p. 125], Rolf E. Hummel, Springer, 2004&lt;/ref&gt;
{{clear}}

==-3rd Century==
[[Image:Sea island survey.jpg|right|thumb|250px|Sea Island survey diagram 窥望海岛之图 was first written of by the Chinese mathematician Liu Hui during the Three Kingdoms era (220–280 CE). Credit: Liu Hui.{{tlx|free media}}]]
[[Image:Mexico.Mex.Teotihuacan.PyramidMoon.01.jpg|thumb|right|360px|The Pyramid of the Moon is one of several monuments built in Teotihuacán apparently in the 3rd century BC. Credit: Hajor.{{tlx|free media}}]]
A Sea Island survey diagram shown on the right was first written of by the Chinese mathematician Liu Hui during the Three Kingdoms era (220–280 CE).

&quot;Jastorf (La Tène) culture [3rd to 1st century BC] with bronze and iron technology. Rich building evidence in downtown Bremen.&quot;&lt;ref name=Heinsohn2017/&gt;

&quot;Many of the coins found at Heracleion date to the time of King Ptolemy II, who ruled Egypt from 283 to 246 B.C.&quot;&lt;ref name=Moss/&gt;
{{clear}}

==-4th Century==
[[Image:Dioptra-principe2.jpg|thumb|right|300px|This is a diagram that shows the operational principles of the dioptra. Credit: [[commons:User:Nerijp|Nerijp]].{{tlx|fairuse}}]]
[[Image:PtolemyWithElephants.jpg|thumb|right|300px|Ptolemaic coins are from the submerged Heracleion. Credit: [[c:user:World Imaging|World Imaging]].]]
A 3rd&amp;nbsp;century BCE '''dioptra''' in the image at the right is an astronomical and surveying instrument, a sighting tube if fitted with protractors, it could be used to measure angles.

Numerous finds from the [Thonis-Heracleion] site have indicated that the city's period of major activity ran from the 6th to the 4th century BC,&lt;ref name=&quot;Lost city of Heracleion gives up its secrets&quot;&gt;{{cite web |url= https://www.telegraph.co.uk/news/earth/environment/archaeology/10022628/Lost-city-of-Heracleion-gives-up-its-secrets.html |title=Lost city of Heracleion gives up its secrets |work=The Telegraph |accessdate = 2013-06-06}}&lt;/ref&gt; with finds of pottery and coins appearing to stop at the end of the 2nd century BC.&lt;ref name=LostCities6/&gt; Goddio's finds have also included incomplete statues of the god Serapis and the queen Arsinoe II.&lt;ref name=&quot;Telegraph-Goddio&quot;&gt;{{cite web|last1=Sooke|first1=Alastair|title=Sunken Cities: the man who found Atlantis|url=https://www.telegraph.co.uk/art/what-to-see/sunken-cities-the-man-who-found-atlantis/|website=The Telegraph|accessdate=22 September 2016}}&lt;/ref&gt; No more than 5% of the city's area was explored by the archaeologist.&lt;ref name=LostCities6/&gt;
{{clear}}

==-5th Century==
[[Image:Edzna, Five-Story Building (14364891152).jpg|right|thumb|300px|The Gran Acropolis is in Edzná. Credit: [https://www.flickr.com/people/67769030@N07 Arian Zwegers from Brussels, Belgium].{{tlx|free media}}]]
[[Image:The Parthenon in Athens.jpg|right|thumb|300px|The Parthenon, Athens Greece, photo taken in 1978. Credit: [http://flickr.com/photos/68686051@N00 Steve Swayne] .{{tlx|free media}}]]
Edzná is a Mayan archaeological site in the north of the Mexican state of Campeche.

The Gran Acropolis is on the right.

Edzná was already inhabited in 400 BC (2400 b2k).

Apparently construction on the Parthenon is dated to have begun in 447 BC (2447 b2k) and was completed in 438 BC (2438 b2k) although decoration continued until 432 BC (2432 b2k).

&quot;The fifth-century B.C. historian Herodotus writes that the Temple of Heracles [in Thonis (Heracleion)] was a refuge for runaway slaves. Herodotus notes that if a slave took refuge there and had the “sacred marks” set on them it would not be lawful for the slave’s original owner to claim them.&quot;&lt;ref name=Moss/&gt;
{{clear}}

==Subatlantic period==
[[Image:Mexico.Mich.Tzintzuntzan.01.jpg|right|thumb|300px|The archaeological site of Tzintzuntzan is the capital of the Tarascan state. Credit: [[c:user:Hajor~commonswiki|Hajor]].{{tlx|free media}}]]
The &quot;calibration of radiocarbon dates at approximately 2500-2450 BP [2500-2450 b2k] is problematic due to a &quot;plateau&quot; (known as the &quot;Hallstatt-plateau&quot;) in the calibration curve [...] A decrease in solar activity caused an increase in production of &lt;sup&gt;14&lt;/sup&gt;C, and thus a sharp rise in Δ &lt;sup&gt;14&lt;/sup&gt;C, beginning at approximately 850 cal (calendar years) BC [...] Between approximately 760 and 420 cal BC (corresponding to 2500-2425 BP [2500-2425 b2k]), the concentration of &lt;sup&gt;14&lt;/sup&gt;C returned to &quot;normal&quot; values.&quot;&lt;ref name=Speranza&gt;{{ cite journal
|author=A. Speranza
|author2=J. van der Plicht
|author3=B. van Geel
|title=Improving the time control of the Subboreal/Subatlantic transition in a Czech peat sequence by &lt;sup&gt;14&lt;/sup&gt;C wiggle-matching
|journal=Quaternary Science Reviews
|month=November
|year=2000
|volume=19
|issue=16
|pages=1589-1604
|url=http://www.researchgate.net/publication/30494985_Improving_the_time_control_of_the_SubborealSubatlantic_transition_in_a_Czech_peat_sequence_by_14C_wiggle-matching/file/60b7d51c350cf2efa0.pdf
|arxiv=
|bibcode=
|doi=10.1016/S0277-3791(99)00108-0
|pmid=
|accessdate=2014-11-04 }}&lt;/ref&gt;

&quot;The main discontinuity in the climatic condition during the Bronze Age and Iron Age transition can be identified in the boundary from Subatlantic to Subboreal (2800-2500 BP; 996/914-766/551 2σ cal. BC). Such period “has globally been identified as a time of marked climatic change. Stratigraphical, paleobotanical and archaeological evidence point to a change from a dry and warm to a more humid and cool climate in central and northwestern Europe” (Tinner et al. 2003). The climatic deterioration which characterizes this chronological range is directly responsible of the plateau in the calibration curve between 760 and 420 BC (2500-2425 BP) (see chapter 4.3.2.1). The climatic oscillation around 2700 BP (896/813 2σ cal. BC) has been detected worldwide. Van Geel et al. (1996, 1998) and Speranza et al. (2002) found an abrupt shift around 850 BC in changing species composition of peat-forming mosses in European Holocene raised bog deposits. The change was from mosses preferring warm conditions to those preferring colder and wetter environments. Archaeological evidence supports such a change. Bronze Age settlements located in the Netherlands were suddenly abandoned after a long period of occupation which last around one millennium (Dergachev et al. 2004). Other studies confirmed the climatic discontinuity; Schilman et al. (2001) studied δ18O and δ13C in deposits from the southeastern Mediterranean, off Israel, and recognized the presence of two humid events in the time ranges of 3500-3000 BP (1884/1772-1263/1215 2σ cal. BC) and 1700-1000 BP (332/389-1016/1030 2σ cal. AD) and a period of arid conditions between 3000 and 1700 BP (1263/1215 2σ cal. BC- 332/389 2σ cal. AD). Barber and Langdon (2001) identified three main long climatic deteriorations 2900-2830 BP (1119/1037-1012/934 2σ cal. BC), 2630-2590 BP (810/797-801/788 2σ cal. BC) and 1550-1400 BP (430/549-637/658 2σ cal. AD) through the analysis of plant macrofossils in a peat deposit of Walton Moss located in Northern England and comparing such data with a temperature reconstruction based on chironomids in the sediment of a nearby lake.&quot;&lt;ref name=Capuzzo/&gt;

&quot;The first (purely typological) studies on Early Bronze Age (EBA) assemblages in the Jordan Valley settled on the turn of the 4th/3rd millennium BC [mark] the beginnings of the earliest Bronze Age culture (Albright 1932; Mallon 1932).&quot;&lt;ref name=Bourke&gt;{{ cite journal
|author=S Bourke
|author2=U Zoppi
|author3=J Meadows
|author4=Q Hua
|author5=S Gibbins
|title=The beginning of the Early Bronze Age in the north Jordan Valley: new 14C determinations from Pella in Jordan
|journal=Radiocarbon
|month= January
|year=2009
|volume=51
|issue=3
|pages=905-913
|url=https://www.researchgate.net/profile/Quan_Hua2/publication/260038389_The_Beginning_of_the_Early_Bronze_Age_in_the_North_Jordan_Valley_New_14C_determinations_from_Pella_in_Jordan/links/00b49537d32bf17b4a000000.pdf
|arxiv=
|bibcode=
|doi=10.2458/azu_js_rc.51.3549
|pmid=
|accessdate=2016-10-23 }}&lt;/ref&gt;
{{clear}}

==&quot;Hallstatt disaster&quot;==
[[Image:Hallstatt Plateau IntCal13 calibration curve.png|right|thumb|300px|&quot;Hallstatt disaster&quot; refers to the plateau located in the calibration curve between 760 and 420 cal BC (2500-2425 BP). Credit: Giacomo Capuzzo.{{tlx|fairuse}}]]
&quot;With the term “Hallstatt disaster” the scientific community refers to the plateau located in the calibration curve between 760 and 420 cal BC (2500-2425 BP) [the graph on the right]. The term is due to the chronological analogy to the Hallstatt society which developed in the late Bronze Age and the beginning of Iron Age in the northern part of the Alps (Austria). The flat shape of the calibration curve in this time-span is the result of the decrease, and hence the return to normal values, of the percentage of &lt;sup&gt;14&lt;/sup&gt;C after a period characterized by an increase in the concentration of radiocarbon in the atmosphere, which is mirrored in the calibration curve as a sharp descent between 850 and 760 BC (2700-2450 BP) (Speranza et al. 2000). As asserted by many authors (Van Geel et al. 1996; Van Geel et al. 1998; Tinner et al. 2003; Dergachev et al. 2004; Van der Plicht et al. 2004; Swindles et al. 2007) the chronological range 850-760 BC is characterized by an abrupt increase of the amount of &lt;sup&gt;14&lt;/sup&gt;C in the atmosphere and it corresponds chronologically to the boundary from Subatlantic to Subboreal (2800-2500 BP), which “has globally been identified as a time of marked climatic change. Stratigraphical, paleobotanical and archaeological evidence point to a change from a dry and warm to a more humid and cool climate in central and northwestern Europe” (Tinner et al. 2003).&quot;&lt;ref name=Capuzzo&gt;{{ cite book
|author=Giacomo Capuzzo
|title=SPACE-TEMPORAL ANALYSIS OF RADIOCARBON EVIDENCE AND ASSOCIATED ARCHAEOLOGICAL RECORD: FROM DANUBE TO EBRO RIVERS AND FROM BRONZE TO IRON AGES
|publisher=UNIVERSITAT AUTÒNOMA DE BARCELONA
|location=BARCELONA, Spain
|year=2014
|volume=
|issue=
|pages=416
|url=https://ddd.uab.cat/pub/tesis/2014/hdl_10803_283401/gc1de1.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2017-10-11 }}&lt;/ref&gt;
{{clear}}

==-6th Century==
[[Image:Kroisos stake Louvre G197.jpg|right|thumb|300px|Croesus at the stake is on Side A from an Attic red-figure amphora, ca. 500–490 BC. Credit: Myson.{{tlx|free media}}]]
[[Image:British Museum gold coin of Croesus.jpg|left|thumb|300px|Gold coin of Croesus, the Lydian, around 550 BC, is from modern Turkey. Credit: [[commons:User:BabelStone|BabelStone]].{{tlx|free media}}]]
[[Image:Alkaios Sappho Staatliche Antikensammlungen 2416 n2.jpg|thumb|center|300px|One of the earliest surviving images of Sappho, from c. 470 BC. Credit: Brygos Painter.]]
The map on the right shows Assyria, Babylonia and Armenia.

Croesus, 595 BC – c. 546 BC, was king of Lydia for apparently 14 years: from 560 BC until his defeat by the Persian king Cyrus the Great in 546 BC, or 547 BC. He issued gold-silver alloy coins as on the left.

Based on the pottery shown in the image on the right, Croesus was burned to death.

&quot;Sappho of Eresos&quot; is shown center holding a lyre and plectrum, and turning toward a bearded man with a lyre partially visible on the left.

A general world-wide use of bronze occurred between 5300 and 2600 b2k.
{{clear}}

==-7th Century==
[[Image:Amulet to ward off plague.jpg|right|thumb|220px|An Amulet to ward off plague is inscribed with a quotation from the Akkadian Erra Epic. Credit: [[commons:User:Fæ|Fæ]].]]
[[Image:Esarhaddon.jpg|right|thumb|250px|Esarhaddon, king of Assyria, has his portrait on a stone stele, dated after 671 BC. Credit: Maur.{{tlx|free media}}]]
[[Image:Black basalt monument of king Esarhaddon. It narrates Esarhaddon's restoration of Babylon. Circa 670 BCE. From Babylon, Mesopotamia, Iraq. The British Museum, London.jpg|left|thumb|250px|Black basalt monument of king Esarhaddon narrates Esarhaddon's restoration of Babylon. Circa 670 BCE. From Babylon, Mesopotamia, Iraq. The British Museum, London. Credit: [[commons:User:Neuroforever|Osama Shukir Muhammed Amin FRCP(Glasg)]].{{tlx|free media}}]]
Amulet in the image on the right dates from 800 BC-612 BC for warding off plague.

Esarhaddon, portrayed on a stone stele, dated after 671 BC on the right, apparently ruled Assyria between 681 – 669 BC.

The black basalt monument of king Esarhaddon on the left narrates Esarhaddon's restoration of Babylon, ca. 670 BCE, from Babylon, Mesopotamia, Iraq, now in The British Museum, London.
{{clear}}

==-8th Century==
[[Image:Nimrud lens British Museum.jpg|thumb|right|250px|This image is a photo of the [[w:Nimrud lens|Nimrud lens]] in the [[w:British museum|British museum]]. Credit: [[commons:User:Geni|Geni]].]]
[[Image:Nimrud lens edge on.png|left|thumb|250px|On edge image of the Nimrud lens shows its plano- (left edge) convex (right edge) shape. Credit: Dimitris Plantzos.{{tlx|fairuse}}]]
[[Image:Prothesis Dipylon Painter A517.jpg|thumb|right|300px|Dipylon Vase of the late Geometric period, or the beginning of the Archaic period, is dated c. 750 BC. Credit: Dipylon Master.{{tlx|free media}}]]
[[Image:Sargon II and dignitary.jpg|thumb|left|250px|Sargon II, King of Assyria is depicted here with a dignitary. Credit: [[commons:User:Jastrow|Jastrow]].{{tlx|free media}}]]
[[Image:Elissa Statut.jpg|center|thumb|250px|This is a statue of Elissa, Queen of Carthage. Credit: [[commons:User:Emnamizouni|Emna Mizouni]].{{tlx|free media}}]]
Nimrud lens is plano-convex, Neo-Assyrian, from the North West Palace, Room AB, dated to 750–710 BC, discovered by Austen Henry Layard at the Assyrian palace of Nimrud, in modern-day Iraq.&lt;ref name=Whitehouse1999&gt;{{ cite book
|author=David Whitehouse
|title=World's oldest telescope?
|url=http://news.bbc.co.uk/1/hi/sci/tech/380186.stm
|date=July 1, 1999
|work=BBC News
|accessdate=May 10, 2008 }}&lt;/ref&gt;

The &quot;&quot;Loupe of Sargon,&quot; a piano-convex rock-crystal lentoid [was] excavated by Layard at Nimrud in the 1850s [on the right].&lt;sup&gt;28&lt;/sup&gt; The object is oval (40 x 35 mm) and of uneven thickness (max. th. 22.5 mm) [centered and including the maximum convexness as shown edge-on in the image on the left. Its focal length has been calculated at 112.5 mm. Its nominal magnification is about 2 x but, owing to its imperfect surface, it would be useless as a tool.&quot;&lt;ref name=Plantzos&gt;{{ cite journal
|author=Dimitris Plantzos
|title=Crystals and Lenses in the Graeco-Roman World
|journal=American Journal of Archaeology
|date=July 1997
|volume=101
|issue=3
|pages=451-64
|url=http://www.hist-arch.uoi.gr/prosopiko/plantzos/Crystals.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2011-10-17 }}&lt;/ref&gt;

This Dipylon Vase on the right is apparently from the late Geometric period, or the beginning of the Archaic period, c. 750 BC (2750 b2k). Prothesis scene is exposure of the dead and mourning.

The low-relief from the L wall of the palace of Sargon II at Dur Sharrukin in Assyria (now Khorsabad in Iraq) is apparently dated to c. 716–713 BC. It is the image on the left.

In the center is an image of the upper portion of a statue of Elissa, Queen of Carthage, apparently from the early 8th century or late ninth century.

&quot;The founding of the Phoenician colony of Toscanos would have come a bit later; its earliest levels (Strata I/II) would date from the years 805–780 cal BC.&lt;sup&gt;6&lt;/sup&gt;&quot;&lt;ref name=Aubet2008/&gt;
{{clear}}

==Subboreal period==

The &quot;period around 850-760 BC, [2850-2760 b2k, is] characterised by a decrease in solar activity and a sharp increase of Δ &lt;sup&gt;14&lt;/sup&gt;C [...] the local vegetation succession, in relation to the changes in atmospheric radiocarbon content, shows additional evidence for solar forcing of climate change at the Subboreal - Subatlantic transition.&quot;&lt;ref name=Speranza/&gt;

==-9th Century==
[[Image:ShalmaneserIII.jpg|thumb|right|300px|King Jehu of the Kingdom of Israel bows before Shalmaneser III of Assyria. Credit: [[w:user:Chaldean|Chaldean]].]]
In 842 BC the Kingdom of Israel and the Phoenician cities sent tributes to Shalmaneser III. The image on the right is from the Black Obelisk of Shalmaneser III and depicts King Jehu of the Kingdom of Israel bowing before Shalmaneser III of Assyria.

Carthage was founded in 814 BC.&lt;ref name=&quot;Moscati&quot;&gt;{{ cite book
|author=Sabatino Moscati
|editor= Sabatino Moscati
|title=Colonization of the Mediterranean, In: ''The Phoenicians''
|url=https://books.google.com/books?id=1EEtmT9Tbj4C
|accessdate=12 February 2013
|date=12 January 2001
|publisher=I.B.Tauris
|isbn=978-1-85043-533-4
|page=48 }}&lt;/ref&gt;

&quot;The recent radiocarbon dates from the earliest levels in Carthage situate the founding of this Tyrian colony in the years 835–800 cal BC,&lt;sup&gt;4﻿﻿&lt;/sup&gt; which coincides with the dates handed down by Flavius Josephus and Timeus for the founding of the city.&quot;&lt;ref name=Aubet2008&gt;{{ cite book
|title=Political and Economic Implications of the New Phoenician Chronologies
|url=http://www.upf.edu/larq/_pdf/AubtCrono.pdf
|publisher=Universidad Pompeu Fabra
|accessdate=24 February 2013
|author=Maria Eugenia Aubet
|pages=179
|date=2008 }}&lt;/ref&gt;

The &quot;dates for the founding of Carthage coincide with dates established for the Phoenician colony of Morro de Mezquitilla, which situate the most ancient occupation levels (Strata B1a and B2) in the years 807–802 cal BC.&lt;sup&gt;5&lt;/sup&gt;&quot;&lt;ref name=Aubet2008/&gt;

The &quot;founding of the first Phoenician colonies at the end of the ninth century coincides with the radio-carbon chronology attributed to the most ancient Phoenician imports recorded in the indigenous Portuguese settlements and in the south of Spain, like Acinipo, Alcáçova de Santarem or Cerro de la Mora.&lt;sup&gt;7&lt;/sup&gt;&quot;&lt;ref name=Aubet2008/&gt;
{{clear}}

==Late Bronze Ages==

The Late Bronze Ages begin about 3550 b2k and end about 2900 b2k.

The Pátzcuaro Basin is &quot;on the Central Mexican Altiplano [19° 36′ N 101° 39′ W 2,033–3,000 meters above sea level (m asl)] [...] The earliest occupation is indicated by maize pollen in lake cores [sometime between 1690 and 940 B.C. (43, 47, 49)].&quot;&lt;ref name=Fisher2003&gt;{{ cite journal
|author=Christopher T. Fisher
|author2=Helen P. Pollard
|author3=Isabel Israde-Alcántara
|author4=Victor H. Garduño-Monroy
|author5=Subir K. Banerjee
|title=A reexamination of human-induced environmental change within the Lake Pátzcuaro Basin, Michoacán, Mexico
|journal=Proceedings of the National Academy of Sciences USA
|month=April
|year=2003
|volume=100
|issue=8
|pages=4957-4962
|url=http://www.pnas.org/content/100/8/4957.long
|arxiv=
|bibcode=
|doi=10.1073/pnas.0630493100
|pmid=
|accessdate=2018-2-25 }}&lt;/ref&gt;

The &quot;abandonment of lakeshore Swiss pile-dwellings has been dated to around 1520 BC [3520 b2k] (Menotti 2001). [Slightly] &quot;later in time episodes of flood events and lake-level highstand at 3100 BP (1415/1311 2σ cal. BC) and 2800 BP (996/914 2σ cal. BC) have been recently detected in the Southern Alps, in the sediment cores extracted from the Lake Ledro, located in the province of Trento (Joannin et al. 2014).&quot;&lt;ref name=Capuzzo&gt;{{ cite book
|author=Giacomo Capuzzo
|title=SPACE-TEMPORAL ANALYSIS OF RADIOCARBON EVIDENCE AND ASSOCIATED ARCHAEOLOGICAL RECORD: FROM DANUBE TO EBRO RIVERS AND FROM BRONZE TO IRON AGES
|publisher=UNIVERSITAT AUTÒNOMA DE BARCELONA
|location=BARCELONA, Spain
|year=2014
|volume=
|issue=
|pages=416
|url=https://ddd.uab.cat/pub/tesis/2014/hdl_10803_283401/gc1de1.pdf
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2017-10-11 }}&lt;/ref&gt;

Radiocarbon &quot;data indicate that the New Kingdom of Egypt started between 1570 and 1544 B.C.E [3570 - 3544 b2k].&quot;&lt;ref name=Ramsey&gt;{{ cite journal
|author=Christopher Bronk Ramsey
|author2=Michael W. Dee
|author3=Joanne M. Rowland
|author4=Thomas F. G. Higham
|author5=Stephen A. Harris
|author6=Fiona Brock
|author7=Anita Quiles
|author8=Eva M. Wild
|author9=Ezra S. Marcus
|author10=Andrew J. Shortland
|title=Radiocarbon-Based Chronology for Dynastic Egypt
|journal=Science
|month=18 June
|year=2010
|volume=328
|issue=5985
|pages=1554-1557
|url=http://science.sciencemag.org/content/328/5985/1554
|arxiv=
|bibcode=
|doi=10.1126/science.1189395
|pmid=
|accessdate=2017-10-11 }}&lt;/ref&gt;
{{clear}}

==-10th Century==
[[Image:Psusennes I mask.jpg|thumb|right|250px|Gold burial mask is for Pharaoh Psusennes I, discovered 1940 by Pierre Montet. Credit: [[commons:User:Locutus_Borg|José-Manuel Benito]].{{tlx|free media}}]]
[[Image:Mask_of_Amenemope1.jpg|left|thumb|250px|Burial mask is for Pharaoh Amenemope, currently in the Cairo Museum. Credit: [http://www.flickr.com/people/10647023@N04/ tutincommon (John Campana)].{{tlx|free media}}]]
The 10th century BC is equivalent to 3,000 to 2,900 b2k.

On the right is the burial mask for the Pharaoh Psusennes I exhumed from Tomb III at Tanis (Nile delta). The material used is gold (different pieces were assembled using nails). The eyebrows and eye shadows are lapis lazuli. The eyes are glass paste. There is a cobra on the forehead. The ritual beard is braided to symbolize the death of the sovereign. It is kept in the Egyptian Museum of Cairo. Dynasty is the XXI, dated to c. 1039 BC-990 BC.

In the left image is the burial mask for pharaoh Amenemope of the 21&lt;sup&gt;st&lt;/sup&gt; Dynasty of Egypt, dated to 1001 – 992 BC or 993 – 984 BC.
{{clear}}

==-11th Century==

==-12th Century==
[[Image:PtolemyVIIIStela.jpg|thumb|right|300px|The stelae of Ptolemy VIII is from the temple of Heracleion. Credit: [[c:user:World Imaging|World Imaging]].]]
Thongs-Heracleion's legendary beginnings go back to as early as the 12th century BC, and it is mentioned by ancient Greek historians. Its importance grew particularly during the waning days of the Pharaohs.&lt;ref name=Alberge&gt;{{cite web|author=Dalya Alberge|title=Ancient Egyptian underwater treasures to be exhibited for the first time|url=https://www.theguardian.com/science/2015/aug/02/ancient-egyptian-artefacts-thonis-heracleion-canopus-paris|date=2 August 2015|newspaper=The Guardian}}&lt;/ref&gt;

&quot;Until very recently the site had been known only from a few literary and epigraphic sources, one of which interestingly mentions the site as an emporion, just like Naukratis.&quot;&lt;ref name=naukratis&gt;
[https://www.britishmuseum.org/research/online_research_catalogues/ng/naukratis_greeks_in_egypt/introduction/naukratis_a_city_and_port.aspx Naukratis: a city and trading port in Egypt], British Museum&lt;/ref&gt;

The city was mentioned by the ancient historians Diodorus (1.19.4) and Strabo (17.1.16). Herodotus was told that Thonis was the warden of the Canopic mouth of the Nile: Thonis arrested Alexander (Paris), the son of Priam, because Alexander had abducted Helen of Troy and taken much wealth.&lt;ref name=naukratis /&gt;&lt;ref&gt;Herodotus, ''Histories'', 2.113-115&lt;/ref&gt;

Heracleion is also mentioned in the twin steles of the Decree of Nectanebo I (the first of which is known as the 'Stele of Naukratis'), which specify that one tenth of the taxes on imports passing through the town of Thonis/Herakleion were to be given to the sanctuary of Neith of Sais.&lt;ref name=naukratis /&gt; The city is also mentioned in the Decree of Canopus honoring Pharaoh Ptolemy III, which describes donations, sacrifices and a procession on water.&lt;ref name=Alberge/&gt;
{{clear}}

==Middle Bronze Ages==
[[Image:Fresco of a fisherman, Akrotiri, Greece.jpg|right|thumb|250px|Fresco of ''The Fisherman'' is from Akrotiri, Santorini, Greece at a height of 1.10 m. Credit: Yann Forget.{{tlx|free media}}]]
The Middle Bronze Ages begin about 4100 b2k and end about 3550 b2k.

''The Fisherman'' is a Minoan Bronze Age fresco from Akrotiri, on the Aegean island of Santorini (classically Thera), dated to the Neo-Palatial period (c. 1640–1600 BC). The settlement of Akrotiri was buried in volcanic ash (dated by radiocarbon dating to c. 1627 BC [c. 3626 b2k]) by the Minoan eruption on the island, which preserved many Minoan frescoes like this.

High precision radiocarbon dating of 18 samples from Jericho, including six samples of carbonized cereal from the burnt stratum, gave the age of the strata as 1562 BC, with a margin of error of 38 years [3562 ± 38 b2k].&lt;ref name=Bruins&gt;{{ cite journal
|author=Hendrik Bruins
|author2=Johannes van der Plicht
|title=Tell-es-Sultan (Jericho): Radiocarbon results of short-lived cereal and multiyear charcoal samples from the end of the Middle Bronze Age
|journal=Radiocarbon
|volume=37
|issue=2
|year=1995
|pages=213-220
|url=https://journals.uair.arizona.edu/index.php/radiocarbon/article/viewFile/1666/1670
|accessdate=2017-10-11 }}&lt;/ref&gt;
{{clear}}

==Early Bronze Ages==

The Early Bronze Ages began about 5300 b2k and end about 4100 b2k.

A logboat from Ireland (Inch Abbey, Co. Down) was dendrochronology dated to 4140 b2k.&lt;ref name=Lanting&gt;{{ cite journal
|author=J.N. Lanting
|title=DATES FOR ORIGIN AND DIFFUSION OF THE EUROPEAN LOGBOAT
|journal=Palaeohistoria
|year=2015
|volume=57
|issue=
|pages=627-650
|url=http://ugp.rug.nl/Palaeohistoria/article/download/25107/22563
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2017-10-13 }}&lt;/ref&gt;

A logboat made from alder from Denmark (Verup l) designated K-4098B was radiocarbon dated to 4220 ± 75 b2k.&lt;ref name=Lanting/&gt;

A logboat from Ireland (Ballygowan, Co. AmJagh) designated GrN-20550 was radiocarbon dated to 4660 ± 40 b2k.&lt;ref name=Lanting/&gt;

&quot;In the Chalcolithic/earliest Bronze Age I period (c. 4500±3000 cal BC), copper was mined in open galleries from the massive brown sandstone deposit, which consisted of thick layers of the copper carbonate malachite and chalcocite, a copper sulphide.&quot;&lt;ref name=Ottaway&gt;{{ cite journal
|author=B.S. Ottaway
|title=Innovation, production and specialization in early prehistoric copper metallurgy
|journal=European Journal of Archaeology
|year=2001
|volume=4
|issue=1
|pages=87-112
|url=http://www.tandfonline.com/doi/abs/10.1179/eja.2001.4.1.87
|arxiv=
|bibcode=
|doi=10.1179/eja.2001.4.1.87
|pmid=
|accessdate=2016-10-23 }}&lt;/ref&gt;

The &quot;Atlantic period [is] 4.6–6 ka [4,600-6,000 b2k].&quot;&lt;ref name=Karabanov&gt;{{ cite journal
|author=E.B. Karabanov
|author2=A.A. Prokopenko
|author3=D.F. Williams
|author4=G.K. Khursevich
|title=A new record of Holocene climate change from the bottom sediments of Lake Baikal
|journal=Palaeogeography, Palaeoclimatology, Palaeoecology
|date=March 2000
|volume=156
|issue=3-4
|pages=211–24
|url=http://www.sciencedirect.com/science/article/pii/S0031018299001418
|arxiv=
|bibcode=
|doi=10.1016/S0031-182(99)00141-8
|pmid=
|accessdate=2014-11-04 }}&lt;/ref&gt;
{{clear}}

==-50th Century==
[[Image:Dolmen Guadalperal Verano 2019.jpg|right|thumb|300px|Due to the low water level of the Valdecañas Reservoir, the Doldal de Guadalperal is fully visible. Credit: [[c:user:Pleonr|Pleonr]].{{tlx|free media}}]]
The &quot;Dolmen of Guadalperal [dubbed] the “Spanish Stonehenge,” the monument is a circle of more than 100 standing rocks dating back to 7,000 years ago.&quot;&lt;ref name=Patel&gt;{{ cite web
|author=Kasha Patel
|title=Drought Reveals Lost “Spanish Stonehenge”
|publisher=NASA
|location=Washington, DC USA
|date=19 September 2019
|url=https://earthobservatory.nasa.gov/images/145619/drought-reveals-lost-spanish-stonehenge
|accessdate=19 September 2019 }}&lt;/ref&gt;

&quot;Originally constructed from granite that was likely transported from miles away, the stones are porous and already falling over, cracking, and eroding.&quot;&lt;ref name=Patel/&gt;

The &quot;dolmen’s most notable feature is a large stone that marks its entrance and features a squiggly line. Some believe it is a map of the Tagus River, which would make it one of the oldest maps on Earth.&quot;&lt;ref name=Patel/&gt;
{{clear}}

==Atlantic==
[[Image:Hieroglyphics found at El-Khawy in Egypt show two storks, back to back.jpg|right|thumb|300px|Hieroglyphics found at El-Khawy in Egypt show two storks, back to back, with an ibis between them (left), as well as a bull's head (right). Credit: John Darnell, Yale University.{{tlx|fairuse}}]]
[[Image:A little elephant is shown inside an adult elephant.jpg|left|thumb|250px|A little elephant is shown inside an adult elephant, an indication that the animal is pregnant. Credit: John Darnell, Yale University.{{tlx|fairuse}}]]
The &quot;Atlantic period [is from] 4.6–6 ka [6,000 to 4,600 b2k].&quot;&lt;ref name=Karabanov/&gt;

&quot;This newly discovered rock art site of El-Khawy preserves some of the earliest — and largest — signs from the formative stages of the hieroglyphic script [such as the back-to-back storks in the image on the right dating back around 5,200 years] and provides evidence for how the ancient Egyptians invented their unique writing system.&quot;&lt;ref name=Darnell&gt;{{ cite book
|author=John Darnell
|title=5,000-Year-Old 'Billboard' of Hieroglyphs Contains a Cosmic Message
|publisher=Live Science
|location=
|date=22 June 2017
|url=https://www.livescience.com/59588-billboard-of-hieroglyphs-contains-cosmic-message.html
|accessdate=2017-06-25 }}&lt;/ref&gt;

Another &quot;carving, [on the left, shows] a herd of elephants, created sometime between 4000 B.C. and 3500 B.C. One of the adult elephants in the scene was drawn with a little elephant inside its body [in the image on the right] — an incredibly rare way of representing a pregnant female animal.&quot;&lt;ref name=Darnell/&gt;

The &quot;reign of Djoser in the Old Kingdom started between 2691 and 2625 B.C.E.&quot;&lt;ref name=Ramsey/&gt;

&quot;The last remains of the American ice sheet disappeared about 6000 years ago [6,000 b2k]&quot;.&lt;ref name=Dansgaard&gt;{{ cite book
|author=Willi Dansgaard
|title=Frozen Annals Greenland Ice Cap Research
|publisher=Niels Bohr Institute
|location=Copenhagen, Denmark
|date=2005
|editor=The Department of Geophysics of The Niels Bohr Institute for Astronomy Physics and Geophysics at The University of Copenhagen Denmark
|pages=123
|url=http://www.iceandclimate.nbi.ku.dk/publications/FrozenAnnals.pdf/
|arxiv=
|bibcode=
|doi=
|pmid=
|isbn=87-990078-0-0
|accessdate=2014-10-05 }}&lt;/ref&gt;

Beginning with the temperatures, as derivable from Greenland ice core data, it is possible to define an 'Early' or 'Pre-Atlantic' period at around 8040 BC, where the &lt;sup&gt;18&lt;/sup&gt;O isotope line remains above 33 ppm in the combined curve after Rasmussen et al. (2006),&lt;ref&gt;S.O. Rasmussen, B.M. Vinther, H.B. Clausen, and K.K. Andersen. Greenland Ice Core Chronology 2005 (GICC05) Early Holocene section. IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series # 2006-119. NOAA/NCDC Paleoclimatology Program, Boulder CO, USA.&lt;/ref&gt; which then would end at the well-known 6.2 ka BC (8.2 ka calBP)-cold-event.

&quot;The Atlantic is equivalent to Pollen Zone VII.&quot;&lt;ref name=Botteville&gt;{{ cite book
|author=[[w:User:Botteville|Botteville]]
|title=Atlantic (period)
|publisher=Wikimedia Foundation, Inc
|location=San Francisco, California
|date=19 February 2006
|url=https://en.wikipedia.org/wiki/Atlantic_(period)
|accessdate=15 July 2018 }}&lt;/ref&gt;
{{clear}}

==Boreal transition==

&quot;In some cores a narrow band of clay interrupts the organic muds, at the horizon of the Boreal Atlantic transition.&quot;&lt;ref name=Franks&gt;{{ cite journal
|author=J. W. Franks
|author2=W. Pennington
|title=The Late-Glacial and Post-Glacial Deposits of the Esthwaite Basin, North Lancashire
|journal=New Phytologist
|month=April
|year=1961
|volume=60
|issue=1
|pages=27-42
|url=http://onlinelibrary.wiley.com/store/10.1111/j.1469-8137.1961.tb06237.x/asset/j.1469-8137.1961.tb06237.x.pdf;jsessionid=EB6966DF0A2FBCC3534CCD6A6413808D.f02t01?v=1&amp;t=i23es9k1&amp;s=e619673cf5bc8be51450a303a914df03f8cba94d
|arxiv=
|bibcode=
|doi=
|pmid=
|accessdate=2014-11-04 }}&lt;/ref&gt;

==Chalcolithic==

The '''copper age history''' period began from 6990 b2k.

The Chalcolithic is often referred to as the Copper Age.

The &quot;oldest securely dated evidence of copper making, from 7,000 years ago [6990 b2k], at the archaeological site of Belovode, Serbia.&quot;&lt;ref name=Radivojevic&gt;{{ cite book
|author=Miljana Radivojevic
|author2=Thilo Rehren
|title=Serbian site may have hosted first copper makers
|publisher=UCL Institute of Archaeology
|location=London, England
|date=23 September 2010
|url=http://www.ucl.ac.uk/archaeology/calendar/articles/20100924
|accessdate=2015-01-18 }}&lt;/ref&gt;

==Neolithic==

The base of the Neolithic is approximated to 12,200 b2k. The transition to the Chalcolithic is between 6,500 and 4,000 b2k.

==Mesolithic==
[[Image:OpgravingStevoort.jpg|right|thumb|300px|Mesolithic artefacts (most Wommersom quartzite) were found during excavation in Stevoort, 2008 (Collection Prehistoric Archeology K.U. Leuven). Credit: [[c:user:Vaneiles|Vaneiles]].{{tlx|free media}}]]
[[Image:Большой шигирский идол.jpg|thumb|200px|upright|left|The Shigir Idol is displayed. Credit: [[c:user:Фальшивомонетчик|Владислав Фальшивомонетчик]].{{tlx|free media}}]]
The '''mesolithic''' period dates from around 13,000 to 8,500 b2k.

By the time of Vere Gordon Childe's work, ''The Dawn of Europe'' (1947), which affirms the Mesolithic, sufficient data had been collected to determine that a transitional period between the Paleolithic and the Neolithic was indeed a useful concept.&lt;ref&gt;{{cite book
|last1=Linder
|first1=F.
|title=Social differentiering i mesolitiska jägar-samlarsamhällen
|date=1997
|publisher=Institutionen för arkeologi och antik historia, Uppsala universitet
|location=Uppsala.
}}&lt;/ref&gt;

The Mesolithic began with the Holocene warm period around 11,660&amp;nbsp;BP and ended with the introduction of farming, the date of which varied in each geographical region. Regions that experienced greater environmental effects as the last glacial period ended have a much more apparent Mesolithic era, lasting millennia.&lt;ref&gt;{{cite journal
|last1=Conneller
|first1=Chantal
|last2=Bayliss
|first2=Alex
|last3=Milner
|first3=Nicky
|last4=Taylor
|first4=Barry
|title=The Resettlement of the British Landscape: Towards a chronology of Early Mesolithic lithic assemblage types
|journal=Internet Archaeology
|date=2016
|volume=42
|doi=10.11141/ia.42.12
}}&lt;/ref&gt;
{{clear}}

==Paleolithic==
[[Image:Cave of Altamira and Paleolithic Cave Art of Northern Spain-110113.jpg|thumb|right|300px|Cave of Altamira and Paleolithic Cave Art of Northern Spain are shown. Credit: Yvon Fruneau, photographer.]]
The '''paleolithic''' period dates from around 2.6 x 10&lt;sup&gt;6&lt;/sup&gt; b2k to the end of the Pleistocene around 12,000 b2k.

The '''Paleolithic''' or '''Palaeolithic''' is a period in human prehistory distinguished by the original development of stone tools that covers c. 95% of human technological prehistory.&lt;ref&gt;{{cite book
|last1=Christian
|first1=David
|title=Big History: Between Nothing and Everything
|date=2014
|publisher=McGraw Hill Education
|location=New York
|page=93
|accessdate=
}}&lt;/ref&gt; It extends from the earliest known use of stone tools by hominins c. 3.3&amp;nbsp;million years ago, to the end of the Pleistocene c. 11,650&amp;nbsp;cal&amp;nbsp;BP.&lt;ref name=Thoth&amp;Schick&gt;Toth, Nicholas; Schick, Kathy (2007). {{cite journal|url=http://www.springerlink.com/content/u68378621542472j|title=Handbook of Paleoanthropology | doi=10.1007/978-3-540-33761-4_64 | journal=Handbook of Paleoanthropology|pages=1943–1963}} In Henke, H.C. Winfried; Hardt, Thorolf; Tatersall, Ian. ''Handbook of Paleoanthropology''. Volume 3. Berlin; Heidelberg; New York: Springer-Verlag. p. 1944. (PRINT: {{ISBN|978-3-540-32474-4}} ONLINE: {{ISBN|978-3-540-33761-4}})&lt;/ref&gt;

{| class=&quot;wikitable&quot; style=&quot;margin:0.5em auto; text-align:center;&quot;
|+ Currently agreed upon classifications as Paleolithic geoclimatic episodes&lt;ref&gt;Gamble, Clive (1990), El poblamiento Paleolítico de Europa, Barcelona: Editorial Crítica. {{ISBN|84-7423-445-X}}.&lt;/ref&gt;
|- valign=&quot;top&quot;
!Age &lt;br /&gt;(before)!!North America!!England (Atlantic Europe)!!Maghreb!!Italy!!Central Europe
|-
|| 10,000 years ||Flandrian interglacial ||Flandriense ||''Mellahiense'' ||''Versiliense'' ||Flandrian interglacial
|-
| 80,000 years ||'''Wisconsin''' || '''Devensiense'''||Regresión || Regresión||'''Wisconsin Stage'''
|-
|| 140,000 years ||Sangamoniense ||Ipswichiense ||''Ouljiense'' ||''Tirreniense II y III'' ||Eemian Stage
|-
|| 200,000 years ||'''Illinois''' ||'''Wolstoniense''' ||Regresión ||Regresión ||'''Wolstonian Stage'''
|-
|| 450,000 years ||Yarmouthiense ||Hoxniense ||''Anfatiense'' ||''Tirreniense I'' ||Hoxnian Stage
|-
| 580,000 years ||'''Kansas''' ||'''Angliense''' ||Regresión ||Regresión ||'''Kansan Stage'''
|-
|| 750,000 years ||Aftoniense ||Cromeriense ||''Maarifiense'' ||''Siciliense'' ||Cromerian Complex
|-
|| 1,100,000 years ||'''Nebraska''' ||'''Beestoniense''' ||Regresión ||Regresión ||'''Beestonian stage'''
|-
|| 1,400,000 years ||interglaciar ||Ludhamiense ||''Messaudiense'' ||''Calabriense'' ||Donau-Günz
|}
{{clear}}

==Marine Isotope Stage 3==
{{Gallery
|title=Stone hand axes found at Lynford Quarry
|width=160
|height=100
|lines=3
|align=left
|Image:Lynford Hand axe-Discoid.jpg|
alt1=|Discoid hand axe
|Image:Lynford Hand axe-Cordiform.jpg|
alt2=|Cordiform hand axe
|Image:Lynford Hand axe-Amydaloid.jpg|
alt3=|Amygdaloid hand axe }}

In archaeology, a '''bout-coupé''' is a type of handaxe that constituted part of the Neanderthal Mousterian industry of the Middle Palaeolithic. The handaxes are bifacially-worked and in the shape of a rounded triangle. They are only found in Britain in the Marine Isotope Stage 3 (MIS 3) interglacial between 59,000 and 41,000 years BP, and are therefore considered a unique diagnostic variant.&lt;ref&gt;{{cite book
|title=The British Palaeolithic: Human Societies at the Edge of the Pleistocene World
|first1=Paul
|last1=Pettitt
|first2=Mark
|last2=White
|publisher=Routledge
|date=2012
|page=349
|location=Abingdon, UK
|isbn=978-0-415-67455-3 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal
|url=http://onlinelibrary.wiley.com/doi/10.1111/1468-0092.00152/abstract
|author=White, Mark J
|author2=Jacobi, Roger M
|title=Two Sides to Every Story: Bout Coupé Handaxes Revisited
|date=May 2002
|publisher=Wiley Online Library
|journal=Oxford Journal of Archaeology
|volume=21
|issue=2
|doi=10.1111/1468-0092.00152
|pages=109–133 }}&lt;/ref&gt;

Lynford Quarry is the location of a well-preserved ''in-situ'' Middle Palaeolithic open-air site near Mundford, Norfolk.&lt;ref name=Lynford&gt;{{Cite book|title=Lynford Quarry, Mundford, Norfolk|publisher=English Heritage
|url=http://webarchive.nationalarchives.gov.uk/20060214231123/http://www.english-heritage.org.uk/server/show/conWebDoc.3892
|date=30 May 2003|accessdate=17 August 2014}}&lt;/ref&gt;

==Ebersdorf Stadial==
[[Image:Australia Genetic Archaeological data Aboriginal populations.jpg|right|thumb|300px|This map of Australia combines genetic and archaeological data to show the movements of Aboriginal populations going back tens of thousands of years. Credit: Alan Cooper and Ray Tobler, University of Adelaide.{{tlx|fairuse}}]]
&quot;Genetics suggests Neanderthal numbers dropped sharply around 50,000 years ago. This coincides with a sudden cold snap, hinting climate struck the first blow.&quot;&lt;ref name=Brahic&gt;{{ cite journal
|author=Catherine Brahic
|title=Human exodus may have reached China 100,000 years ago
|journal=New Scientist
|location=
|month=08 August
|year=2014
|url=http://www.newscientist.com/article/mg22329813.000-human-exodus-may-have-reached-china-100000-years-ago.html#.U-_PEShOSlI
|accessdate=2014-08-16 }}&lt;/ref&gt;

&quot;The first humans probably reached Australia around 50,000 years ago, which is the age of the oldest human skeletons and tools found.&quot;&lt;ref name=Cooper&gt;{{ cite book
|author=Alan Cooper
|title=How the Aborigenes came to Australia
|publisher=Q-Magazine
|location=
|date=
|url=http://www.q-mag.org/how-the-aborigenes-came-to-australia.html
|accessdate=2017-05-29 }}&lt;/ref&gt;

All &quot;the Aborigines likely descend from a single population, which reached the Australian continent 50,000 years ago. Populations then spread rapidly – within 1,500 to 2,000 years – around the east and west coasts of Australia, meeting somewhere in South Australia. Over the following millennia, the population groups remained practically isolated.&quot;&lt;ref name=Cooper/&gt;

&quot;Australia 50,000 years ago was part of the same landmass as New Guinea. So that the first Aborigines could have reached New Guinea by way of South East Asia and then have gone farther to Australia. There, they settled in groups over the whole continent.&quot;&lt;ref name=Cooper/&gt;

Many &quot;groups of Aborigines used similar tools and shared a similar language. If humans did not move, how could tools and languages?&quot;&lt;ref name=Bellwood&gt;{{ cite book
|author=Peter Bellwood
|title=How the Aborigenes came to Australia
|publisher=Q-Magazine
|location=
|date=9 March 2017
|url=http://www.q-mag.org/how-the-aborigenes-came-to-australia.html
|accessdate=2017-05-29 }}&lt;/ref&gt;
{{clear}}

==Marine Isotope Stage 4==
[[Image:Neanderthalguerinnicolas.jpg|left|thumb|300px|Neanderthal skull, Museum d'Anthropologie, campus universitaire d'Irchel, Université de Zurich (Suisse), is imaged. Credit: Guerin Nicolas.{{tlx|fairuse}}]]
&quot;During the Middle Stone Age of Southern Africa, technological and behavioral innovations led to significant changes in the lifeways of modern humans. The glacial episode of Marine Isotope Stage 4, about 57-71,000 years ago, resulted in cooler and drier climatic conditions and the expansion of grassland vegetation. Sea level dropped by as much as 80 meters below its current level. During this period the cultural phase known as the Howieson’s Poort appeared across much of Southern Africa, peaking at about 60-65,000 years ago, and then disappeared. The lithic industry of the Howieson’s Poort is exemplified by changes in technology, such as the use of the punch technique, an increase in the selection of fine-grained silcrete, and the predominance of retouched pieces including backed tools, segments, scrapers and points. Segments are the type fossil of the Howieson’s Poort and represent multi-purpose armatures that were hafted onto wooden spear shafts. The standardized design and refined style of segments convey information about the behavior of their makers and provide insight about group identity. Increasing use of ochre, the presence of engraved ostrich eggshells, and a bone tool industry are associated with these stone artifacts. Also evident is an intensified use of space. Taken together, these behaviors suggest that the Howieson’s Poort represents a clear marker of modern human culture.&quot;&lt;ref name=ROCEEH&gt;{{ cite book
|author=ROCEEH
|title=File:Motm 2010 07 Howiesons Poort.pdf
|publisher=Wikimedia
|location=
|date=1 July 2010
|url=https://commons.wikimedia.org/wiki/File:Motm_2010_07_Howiesons_Poort.pdf
|accessdate=11 July 2018 }}&lt;/ref&gt;

&quot;Using stone tool residue analysis with supporting information from zooarchaeology, we provide evidence that at the Abri du Maras, Ardèche, France, Neanderthals [a skull is imaged on the left from Abri du Maras] were behaviorally flexible at the beginning of MIS 4. Here, Neanderthals exploited a wide range of resources including large mammals, fish, ducks, raptors, rabbits, mushrooms, plants, and wood. Twisted fibers on stone tools provide evidence of making string or cordage.&quot;&lt;ref name=Hardy&gt;{{ cite journal
|author=Bruce L. Hardy
|author2=Marie-Hélène Moncel
|author3=Camille Daujeard
|author4=Paul Fernandes
|author5=Philippe Béarez
|author6=Emmanuel Desclaux
|author7=Maria Gem
|author8=Chacon Navarro
|author9=Simon Puaud
|author10=Rosalia Gallotti
|title=Impossible Neanderthals? Making string, throwing projectiles and catching small game during Marine Isotope Stage 4 (Abri du Maras, France)
|journal=Quaternary Science Reviews
|date=15 December 2013
|volume=82
|issue=12
|pages=23-40
|url=https://www.sciencedirect.com/science/article/pii/S0277379113003788
|arxiv=
|bibcode=
|doi=10.1016/j.quascirev.2013.09.028
|pmid=
|accessdate=12 July 2018 }}&lt;/ref&gt;
{{clear}}

==Hypotheses==
{{main|Hypotheses}}
# Archaeology of Sclandinavia will eventually show it to have been occupied 40,000 b2k.

==See also==
{{div col|colwidth=20em}}
* [[Abstract concept generator]]
* [[Anthropology/Lecture|Anthropology]]
* [[Archeology]] (Portal:Archaeology)
* [[Classical planets]]
* [[Dominant group/Archaeology]]
* [[Dominant group/Culture]]
* [[Dominant group/Metagenome]]
* [[Genetics/Human DNA]]
* [[Genetics/Human teeth]]
* [[Genetics/Mammalogy]]
* [[Geochronology/Archaeology]]
* [[Geochronology/Cenozoic]]
* [[Geochronology/Classics]]
* [[Geochronology/Dates]]
* [[Geochronology/Dendrochronology]]
* [[Geochronology/Dye 3]]
* [[Geochronology/Geomagnetic Polarity Time Scale]]
* [[Geochronology/History]]
* [[Geochronology/Ice cores]]
* [[Geochronology/Ice cores/Black ice]]
* [[Geochronology/Ice cores/Brittle ice]]
* [[Geochronology/Ice cores/Clear ice]]
* [[Geochronology/Ice cores/Firns]]
* [[Geochronology/Ice cores/Sea ice]]
* [[Geochronology/Libyan history]]
* [[Geochronology/Medieval Warm Period]]
* [[Geochronology/Mesozoic]]
* [[Geochronology/Middle Ages]]
* [[Geochronology/Neoglaciations]]
* [[Geochronology/Optically stimulated luminescence]]
* [[Geochronology/Paleontology]]
* [[Geochronology/Paleozoic]]
* [[Geochronology/Palynology]]
* [[Geochronology/Radiocarbon dating]]
* [[Geochronology/Stratigraphy]]
* [[Geochronology/Tephra layers]]
* [[Geochronology/Uranium-thorium dating]]
* [[Hominins]]
* [[Paleanthropology]]
* [[Philosophy/Sciences|Sciences]]
* [[Radiation astronomy/History]]
{{Div col end}}

==References==
{{reflist|2}}

==External links==

&lt;!-- footer templates --&gt;
{{tlx|Anthropology resources}}{{Archaeology resources}}{{tlx|Geology resources}}{{tlx|Humanities resources}}{{Sisterlinks|Archaeology}}{{Sisterlinks|Archeology}}

&lt;!-- categories --&gt;
[[Category:Anthropology/Lectures]]
[[Category:Archaeology/Lectures]]
[[Category:Arts/Lectures]]
[[Category:Cultures/Lectures]]
[[Category:Dominant group/Lectures]]
[[Category:Earth sciences/Lectures]]
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[[Category:Humanities/Lectures]]
[[Category:Literature/Lectures]]</text>
      <sha1>7734ylnichu7kf24i0ba75ketngot8v</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2844978</id>
      <parentid>2844814</parentid>
      <timestamp>2026-11-09T13:03:49Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2844978</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="bnvlji3vi2kot4znb7n7fh3ok1422g4" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20261109.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
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        <id>791434</id>
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      <comment>Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]])</comment>
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    <title>User:Dc.samizdat/Real Euclidean four-dimensional space</title>
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      <text bytes="124184" sha1="3mbc2mr6tans1xmj7q0ewjz1df21e9z" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a 4-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2845020</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124404" sha1="38n613qxacbiy4c80jd1qt67x2q4yy1" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered wherever there is an observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845023</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124390" sha1="px8inlaxqh6tbaiuruaz3ihsme3pud8" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845024</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124403" sha1="pxdhy14vpj7emzoi45j5w4gxgbwak3h" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere perceptually centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845025</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124390" sha1="px8inlaxqh6tbaiuruaz3ihsme3pud8" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845026</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124532" sha1="fq1ezbolfo818adg3d9u6u5jqxz398z" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass move inertially at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction in Euclidean 4-space. Our entire 3-space manifold is moving at the speed of light &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845027</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124535" sha1="drxcwcmjxp004a1h2yssg12nu1ofni1" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass move inertially at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction in Euclidean 4-space. Our entire 3-space manifold is in motion in Euclidean 4-space at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845028</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124521" sha1="qo2kc3hmj9ueg7opqiusk78adedchgg" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845029</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124541" sha1="aucngmamx94m9y36pzs2p5x0tovzwov" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <text bytes="124698" sha1="4extwrxs86ujknguk7okejos9bs91dr" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>4extwrxs86ujknguk7okejos9bs91dr</sha1>
    </revision>
    <revision>
      <id>2845031</id>
      <parentid>2845030</parentid>
      <timestamp>2026-11-09T23:09:59Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="124688" sha1="pipr4suhwelexdvdt7jh8bcbmbnqbfi" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>pipr4suhwelexdvdt7jh8bcbmbnqbfi</sha1>
    </revision>
    <revision>
      <id>2845032</id>
      <parentid>2845031</parentid>
      <timestamp>2026-11-09T23:14:38Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="124688" sha1="hj7vsck2w5ja9s42f7rrel0tycfmnau" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>hj7vsck2w5ja9s42f7rrel0tycfmnau</sha1>
    </revision>
    <revision>
      <id>2845033</id>
      <parentid>2845032</parentid>
      <timestamp>2026-11-09T23:15:41Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="124687" sha1="ssu9qj11bxkmdad1pxnwmqamqemb5yu" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ssu9qj11bxkmdad1pxnwmqamqemb5yu</sha1>
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    <revision>
      <id>2845036</id>
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      <timestamp>2026-11-09T23:29:03Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* An object's motion in space is the local product of its discrete self-reflections */</comment>
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      <text bytes="124693" sha1="ec76c1o9mjl5mc7e38rq7dj925awxfa" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>ec76c1o9mjl5mc7e38rq7dj925awxfa</sha1>
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    <revision>
      <id>2845037</id>
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      <timestamp>2026-11-09T23:35:07Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* An object's motion in space is the local product of its discrete self-reflections */</comment>
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      <text bytes="124862" sha1="hqfntlxqz7mz4con03ododfda6ps7ra" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are evenly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
      <sha1>hqfntlxqz7mz4con03ododfda6ps7ra</sha1>
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    <revision>
      <id>2845070</id>
      <parentid>2845037</parentid>
      <timestamp>2026-11-10T05:44:48Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */</comment>
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      <text bytes="124865" sha1="0noeqtj8evq0wep1kxuclpx8rf4grkf" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845071</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124859" sha1="9d927ffl5u5quzvbgcxo42jl1zijl7r" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845072</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124858" sha1="et7pbj44aoiyx39phpuhr09d0ngabz2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] openly advocated these views and was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers such as Bruno dared to disperse those stars into an infinite Euclidean 3-space, and relinquish our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845073</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124858" sha1="ndp9ibespxdwlxei10o1iizhagz3l7c" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of Bell's theorem, is an omnipresent symmetry property of Euclidean 4-space. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers such as Bruno dared to disperse those stars into an infinite Euclidean 3-space, and relinquish our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845077</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="124918" sha1="t7cctqwdp9pssr7bjhp9vrcj0egky8w" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of the Kochen–Specker theorem, is an omnipresent symmetry property of Euclidean 4-space which does not depend on action at a distance. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers such as Bruno dared to disperse those stars into an infinite Euclidean 3-space, and relinquish our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845078</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124923" sha1="dpjdes71gmnkru63yce517pt2yayqpy" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope. Contextuality, in the sense of the Bell-Kochen–Specker theorem, is an omnipresent symmetry property of Euclidean 4-space which does not depend on action at a distance. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space. 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers such as Bruno dared to disperse those stars into an infinite Euclidean 3-space, and relinquish our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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      <id>2845082</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124932" sha1="ru2dfvktuc9pfwovojcefxm9epq3uhp" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope.{{Efn|Contextuality, in the sense of the Bell–Kochen–Specker theorem, is an omnipresent symmetry property of Euclidean 4-space which does not depend on action at a distance. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space.}} 
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers such as Bruno dared to disperse those stars into an infinite Euclidean 3-space, and relinquish our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845083</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124932" sha1="4a4ho16fb7cs2q4dmit7g7e2ovy6k72" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space.{{Efn|Contextuality, in the sense of the Bell–Kochen–Specker theorem, is an omnipresent symmetry property of Euclidean 4-space which does not depend on action at a distance. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean space.}}  We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers such as Bruno dared to disperse those stars into an infinite Euclidean 3-space, and relinquish our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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    <revision>
      <id>2845086</id>
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        <username>Dc.samizdat</username>
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      <text bytes="124934" sha1="jzquc7bxild5z0v4ic7yhd81nx1qw82" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - November 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.&lt;/blockquote&gt;

== Summary ==
We observe that:
* Physical space is Euclidean with four perpendicular dimensions.
* Atoms are [[W:4-polytope|4-polytopes]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space.
* The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space.
* Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space.
* All objects with mass are in perpetual inertial motion at constant velocity &lt;math&gt;c&lt;/math&gt;, and acceleration only varies their direction of motion in Euclidean 4-space. Our entire 3-space manifold is in motion at the speed of light &lt;math&gt;c&lt;/math&gt;, a motion isomorphic to the passage of proper time. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity &lt;math&gt;c&lt;/math&gt; in Euclidean 4-space.
* Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is &lt;math&gt;c\prime = 2c&lt;/math&gt; measured on a straight line in Euclidean 4-space, notwithstanding the fact that it is &lt;math&gt;c&lt;/math&gt; measured on a geodesic in our non-Euclidean 3-space.
* Time ticks at the same rate everywhere in Euclidean 4-space, notwithstanding the fact that its rate varies with the gravitational field.
* General relativity describes the rotational motion of objects with mass in Euclidean 4-space. The gravitational acceleration of an object is isomorphic to the object's geodesic motion on a 3-sphere undergoing a double rotation in Euclidean 4-space.
* Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space.{{Efn|Contextuality, in the sense of the Bell–Kochen–Specker theorem, is an omnipresent symmetry property of Euclidean 4-space which does not depend on action at a distance. Coxeter group symmetry preserves the intrinsic contextuality and locality of physical actions in Euclidean 4-space.}}  We hypothesize that the proton has the symmetry of the 4-simplex polytope, and the neutron has the symmetry of the 4-orthoplex polytope.
* The universe may properly be viewed as a 4-sphere centered on the observer.

== A theory of the Euclidean cosmos ==
&lt;blockquote&gt;If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.&lt;ref&gt;{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}&lt;/ref&gt;&lt;/blockquote&gt;
The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. 

All objects with mass move inertially through Euclidean 4-space at constant velocity &lt;math&gt;c&lt;/math&gt; as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt; is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space.

A galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. 

Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity &lt;math&gt;c&lt;/math&gt;, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane.

The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity &lt;math&gt;c&lt;/math&gt; in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity &lt;math&gt;c&lt;/math&gt;. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. 

The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together, not some invisible dark matter.

The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity &lt;math&gt;c&lt;/math&gt;, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. 

For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. 

The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. 

A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity &lt;math&gt;c&lt;/math&gt;. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects.

Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. 

Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core.

The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be &quot;heaped up&quot; into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. 

Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker toward its center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball).

The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect.

Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity &lt;math&gt;c&lt;/math&gt; corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity &lt;math&gt;c&lt;/math&gt;, a rotation and a translation.  

Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity &lt;math&gt;c&lt;/math&gt; through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is &lt;math&gt;c\prime = 2c&lt;/math&gt;, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity &lt;math&gt;c&lt;/math&gt;. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. 

This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space.

== Symmetries ==

It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel &amp; Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. 

[[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;math&gt;\mathrm{Q}&lt;/math&gt; denote a rotation, &lt;math&gt;\mathrm{R}&lt;/math&gt; a reflection, &lt;math&gt;\mathrm{T}&lt;/math&gt; a translation, and let &lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt; denote a product of several such transformations, all commutative with one another. Then &lt;math&gt;\mathrm{RT}&lt;/math&gt; is a glide-reflection (in two or three dimensions), &lt;math&gt;\mathrm{QR}&lt;/math&gt; is a rotary-reflection, &lt;math&gt;\mathrm{QT}&lt;/math&gt; is a screw-displacement, and &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r \le n)&lt;/math&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are also expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;.&lt;br&gt;
For &lt;math&gt;(n = 4)&lt;/math&gt; in particular, every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;
If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; or a &lt;math&gt;\mathrm{QT}&lt;/math&gt;, because we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;. By the same principle, we can view any &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; as an isoclinic (equi-angled) &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) for ''one'' of the two rotations (&lt;math&gt;\mathrm{Q}&lt;/math&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;).

As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group &lt;math&gt;SO(4)&lt;/math&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;math&gt;SO(4)&lt;/math&gt; rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics.

== Special relativity describes Euclidean 4-space ==

&lt;blockquote&gt;Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.&lt;ref&gt;{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}&lt;/ref&gt;&lt;/blockquote&gt;

Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.&lt;ref name=Einstein1923&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt;

Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity &lt;math&gt;c&lt;/math&gt;, in some 4-space direction.{{Sfn|Yamashita|2023}}

This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always &lt;math&gt;c&lt;/math&gt;, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn|
...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity &lt;math&gt;c&lt;/math&gt;. In physics as it has been universally understood, observers are not supposed to be able to move at velocity &lt;math&gt;c&lt;/math&gt;. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four.

In the Euclidean relativity alternative view, however, every observer is always moving at velocity &lt;math&gt;c&lt;/math&gt; through the universe, which is real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity &lt;math&gt;c&lt;/math&gt;. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity &lt;math&gt;c&lt;/math&gt;, in the same inertial reference frame. Your proper 4-coordinate reference frame is rotated with respect to another observer's proper 4-coordinate reference frame, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper 4-coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems.

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity &lt;math&gt;c&lt;/math&gt; through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the &quot;speed of light&quot; &lt;math&gt;c&lt;/math&gt;, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity &lt;math&gt;c&lt;/math&gt;, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of light through 3-space. In Euclidean relativity, the invariant &lt;math&gt;c&lt;/math&gt; is the speed of matter through 4-space! The speed of light through 3-space is also perceived as &lt;math&gt;c&lt;/math&gt; by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt;.

Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity &lt;math&gt;c&lt;/math&gt;, or their possession of experimentally undetected dark matter. A more general example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically.{{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers.
...cite Kappraff &amp; Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff &amp; Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff &amp; Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}}

Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could &quot;catch up&quot; with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at &lt;math&gt;c&lt;/math&gt; (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than &lt;math&gt;c&lt;/math&gt;. Euclidean relativity is a revolutionary theory indeed, in which &lt;math&gt;c&lt;/math&gt; cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for &lt;math&gt;\mathbb{R}^4&lt;/math&gt; to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be &lt;math&gt;c\prime = 2c&lt;/math&gt;, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number &lt;math&gt;\sqrt{4}&lt;/math&gt;.

== An object's motion in space is the local product of its discrete self-reflections ==
Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move.

Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a  object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality.

Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space:

&lt;blockquote&gt;Every orthogonal transformation in 4-space is expressible as:&lt;br&gt;
:&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t&lt;/math&gt;&lt;br&gt;
where &lt;math&gt;(2^q + r + t \le 4)&lt;/math&gt;. Every displacement is either a double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, or a screw-displacement &lt;math&gt;\mathrm{QT}&lt;/math&gt; [where the rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt; is a simple rotation, but the &lt;math&gt;\mathrm{QT}&lt;/math&gt; is chiral like a &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;math&gt;\mathrm{QRT}&lt;/math&gt;.&lt;/blockquote&gt;

While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (&lt;math&gt;\mathrm{Q}&lt;/math&gt;), reflection (&lt;math&gt;\mathrm{R}&lt;/math&gt;) and translation (&lt;math&gt;\mathrm{T}&lt;/math&gt;) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate.

Four dimensional Euclidean space can accomodate no such thing as a &lt;math&gt;\mathrm{Q}^2\mathrm{T}&lt;/math&gt;. A translating double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt;, like a translating simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt;, is simply a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt;. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or a simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. In 4-space the orbit of a screw translation &lt;math&gt;\mathrm{QT}&lt;/math&gt; or double rotation &lt;math&gt;\mathrm{Q}^2&lt;/math&gt; lies on a great circle of a hypersphere (a 3-sphere), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. 

A point on the surface of an object rotating in 4-space may be in motion on two great circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not &lt;math&gt;2\pi r&lt;/math&gt;. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. They are greater circles than planar great circles since they equatorially circumnavigate the whole 3-sphere rather than rounding only one 2-sphere within the 3-sphere. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle.

In our universe which is real Euclidean four-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, these motions occur only at the constant velocity &lt;math&gt;c&lt;/math&gt;, and there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a &lt;math&gt;\mathrm{QT}&lt;/math&gt; at velocity &lt;math&gt;c&lt;/math&gt;. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is a common reference frame of all observers.

== Light propagates through 4-space at twice its apparent velocity &lt;math&gt;c&lt;/math&gt;==
Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection &lt;math&gt;\mathrm{R}^4&lt;/math&gt;, which may be seen as a double translation &lt;math&gt;\mathrm{T}^2&lt;/math&gt;, a pure translation via two pairs of parallel reflections without any rotation component &lt;math&gt;\mathrm{Q}&lt;/math&gt;.

Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by &lt;math&gt;\mathrm{QT}&lt;/math&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;math&gt;\mathrm{Q^2}&lt;/math&gt;, such as an atom in its own proper reference frame. A simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; or simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; is a double reflection &lt;math&gt;\mathrm{R^2}&lt;/math&gt;, so a &lt;math&gt;\mathrm{QT}&lt;/math&gt; or &lt;math&gt;\mathrm{Q^2}&lt;/math&gt; is an &lt;math&gt;\mathrm{R^4}&lt;/math&gt;, but not with the same group of reflection angles as a light signal &lt;math&gt;\mathrm{R^4}&lt;/math&gt;. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation &lt;math&gt;\mathrm{T = R^2}&lt;/math&gt; is a double reflection in two parallel hyperplanes, and a rotation &lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt; is a double reflection in two intersecting hyperplanes, as in a &lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt; which is both at once. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation &lt;math&gt;\mathrm{T}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. A double translation &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; is the opposite of a double rotation &lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;, which is stationary but rotates twice as far as the simple rotation &lt;math&gt;\mathrm{Q}&lt;/math&gt; in a &lt;math&gt;\mathrm{QT}&lt;/math&gt;. 

The product of the two translations in a &lt;math&gt;\mathrm{T^2}&lt;/math&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;math&gt;\mathrm{T}&lt;/math&gt; over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;math&gt;\mathrm{T^2}&lt;/math&gt; cannot reposition a 4-polytope the way a &lt;math&gt;\mathrm{QT}&lt;/math&gt; does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn|
...lensing of double translations &lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt; in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}}

== Special relativity is Galilean relativity in Euclidean 4-space ==

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension.

It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity &lt;math&gt;c&lt;/math&gt;, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.)

The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply:

&lt;blockquote&gt;Time ticks at the same rate everywhere in space.&lt;/blockquote&gt;

This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the rates at which clocks run in distinct moving gravitational fields, and special relativity predicts differences in perceived rates at which clocks run in distinct moving inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than &lt;math&gt;c&lt;/math&gt; in any stationary place in 4-space, at any time.

None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does.

The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery.

== General relativity is the geometry of gravity in Euclidean 4-space ==

The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]].

The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames).

Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. 

Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same.

Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. 

Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.

An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity &lt;math&gt;c&lt;/math&gt; on their respective distinct screw translations &lt;math&gt;QT&lt;/math&gt;.

In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation &lt;math&gt;T&lt;/math&gt; through 4-space. The velocity of that vector remains the constant &lt;math&gt;c&lt;/math&gt; throughout; only his direction is changing. A change in the direction of a vector is a rotation &lt;math&gt;Q&lt;/math&gt; of a translation &lt;math&gt;T&lt;/math&gt;: a screw translation &lt;math&gt;QT&lt;/math&gt;.

A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation composed of a discrete number of its characteristic rotational displacement in each full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and discrete number of displacements per revolution.

The general rotation in 4-space is a double rotation &lt;math&gt;Q^2&lt;/math&gt; distinguished by two completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may have to be greater than 360° to close a circular revolution. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we call isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference. 

In 4-space every object with mass moves on its distinct &lt;math&gt;QT&lt;/math&gt; trajectory. Each rotational displacement &lt;math&gt;Q&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement &lt;math&gt;T&lt;/math&gt; is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity &lt;math&gt;c&lt;/math&gt;. Every atom translates at velocity &lt;math&gt;c&lt;/math&gt;, performing a large number &lt;math&gt;c&lt;/math&gt; of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity &lt;math&gt;c&lt;/math&gt;, performing the same large number &lt;math&gt;c&lt;/math&gt; of atomic scale rotational displacements per second.

Although the number of displacements at atomic scale per unit time is the same universal constant &lt;math&gt;c&lt;/math&gt; for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution, and the circumference of the complete revolution, are characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of &lt;math&gt;c&lt;/math&gt; and the two defining angle parameters of the rotation, since they depend on the geometry of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups, which are each their own mystery with a unique conservation law and set of rotations. The unique geometry of each rotation's isocline is that of some distinct star polygon. The geometric relationships between star polygons defined by successive rational numbers are mathematically mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios, which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions, and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator.

The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations, because a translation &lt;math&gt;T&lt;/math&gt; is the distinct simple rotation &lt;math&gt;Q&lt;/math&gt; in which the characteristic angle of rotational displacement is 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing, and inertia is the conservation of this distinct double rotation. The acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity &lt;math&gt;c&lt;/math&gt;. Because the rate of atomic scale displacement events is the universal constant &lt;math&gt;c&lt;/math&gt;, the rate of change of direction (the acceleration of the object) is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry.

General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space: we must imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, visualize its isocline, and model its unique conservation law geometrically as motion on those isoclines in Euclidean 4-space.{{Efn|It will also help to visualize the isocline as a geodesic straight line in spherical 3-dimensional space, on the surface of the 3-sphere.}} That problem is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.&lt;ref&gt;{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton &amp; Company|isbn=978-0393039184}}&lt;/ref&gt; It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space.

A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of &lt;math&gt;c&lt;/math&gt; and the distinct rotation's isocline geometry, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. Our challenge is to visualize this double rotation.

...

== Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space ==

Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. 

Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). 

The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We identify the proton with the rotational symmetry the 4-simplex.

The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other such that their vertices are uniformly distributed on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the singular geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes.

The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the comprehensive 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. If we view the 4-simplex as the largest 4-polytope instead of the smallest, and subdivide it by integral truncation, we find smaller instances of all the other 4-polytopes within it.

The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each atomic nucleus is based on that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, on the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, each occupying exclusively 5 equidistant vertices of the 600 vertices, only 120 distinct atomic elements are possible. Any compound of the atomic number of 4-simplexes constitutes an atomic nucleus of that element, but multiple geometric configurations of each element are possible, depending on which subset of 4-simplex positions is populated. Some of these configurations will have more symmetry than others. We conclude that these alternate configurations potential nuclides of the atomic element, some potentially more stable than others. 

We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. Our conjecture is that each nuclide is a sparse configuration of the 120-cell that contains its atomic number of 4-simplexes and its neutron number of 4-orthoplexes.

The 8-point 4-orthoplex is the easiest 4-polytope to visualize because it consists simply of 4 pairs of opposing vertices on 4 orthogonal axes. It is the [[W:Orthonormal basis|orthonormal basis]] polytope, because its size, location and orientation in 4-space defines a Cartesian (w,x,y,z) coordinate basis, a non-rotating inertial reference frame. Every 4-polytope in the series of six except the 4-simplex is divisible into an integer number of disjoint 4-orthoplexes, each occupying exclusively 8 of the 600 vertices of the 120-cell. Moreover, both the 120-point 600-cell and the 600-point 120-cell are divisible into a larger integer number of non-disjoint 4-orthoplexes which share axes. 

In contrast, the 5-point 4-simplex is the only regular convex 4-polytope with vertices that do not occur in opposing pairs. It is impossible to position two 4-simplexes on the 3-sphere such that they share some, but not all 5 vertices. Therefore the 600-point 120-cell can only be partitioned into 120 disjoint 5-point 4-simplexes in one way, and the smaller 4-polytopes in the series cannot be partitioned into 4-simplexes at all.

The 600-point (300 axis) 120-cell can be partitioned into 75 disjoint (4 axis) 4-orthoplexes. This partitioning can be done in 10 different ways, because two 4-orthoplexes can be positioned so that they share an axis without sharing all 8 vertices. The 120-cell contains 675 distinct 4-orthoplexes, each a distinct orthonormal basis with four orthogonal axes. Each axis occurs in 9 distinct 4-orthoplexes, and is orthogonal to its 27 distinct companion axes in those 4-orthoplexes, and to no other axes of the 120-cell.{{Sfn|Waegell &amp; Aravind|2014}}

Because the 4-simplex has no opposing vertices, its 5 vertices belong to 5 distinct orthonormal reference frames, such that two vertices of the same 4-simplex can never belong to the same 4-orthoplex.

An atomic nucleus of atomic number &lt;math&gt;z&lt;/math&gt; is a compound of &lt;math&gt;z&lt;/math&gt; disjoint 4-simplexes, so it occupies exclusively &lt;math&gt;5 z&lt;/math&gt; vertices of the 120-cell, in some configuration with the 5 vertices of each 4-simplex mutually equidistant from each other on the the 3-sphere. It is also a nuclide of neutron number &lt;math&gt;n&lt;/math&gt;, so it is also a compound of &lt;math&gt;n&lt;/math&gt; distinct 4-orthoplexes, not necessarily disjoint. The 4-simplex compound and 4-orthoplex compound may intersect, subject to the constraints above. The combined compounds form a single sparse configuration of the 120-cell. It is our conjecture that more stable nuclides correspond to sparse configurations with greater symmetry.

....

== Dimensional relativity ==
The formulation of Euclidean relativity based on Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy.

Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]].

By analogy to the surface of the earth, the bounding surface of a spherical region of &lt;math&gt;n&lt;/math&gt;-dimensional Euclidean space is an &lt;math&gt;(n-1)&lt;/math&gt;-sphere, a spherical space of one fewer dimensions than the &lt;math&gt;n&lt;/math&gt;-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical &lt;math&gt;n&lt;/math&gt;-spaces of increasing &lt;math&gt;n&lt;/math&gt;, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating &lt;math&gt;n&lt;/math&gt;-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into &lt;math&gt;n&lt;/math&gt;-dimensional space than others.

== Polycentric spherical relativity ==

An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean &lt;math&gt;n&lt;/math&gt;-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an &lt;math&gt;n&lt;/math&gt;-sphere with themself at the center observing it, perceptually equidistant from all points on its &lt;math&gt;n&lt;/math&gt;-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything.

This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean spaces of any number of dimensions. 

The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space:
&lt;blockquote&gt;The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}&lt;/blockquote&gt;

== Revolutions ==

The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition.

In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity &lt;math&gt;c&lt;/math&gt;, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time.

As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers such as Bruno dared to disperse those stars into an infinite Euclidean 3-space, and relinquish our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space.

Looking still deeper and farther out, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space.

When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean &lt;math&gt;n&lt;/math&gt;-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity &lt;math&gt;c&lt;/math&gt;, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. 

To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity &lt;math&gt;c&lt;/math&gt; in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. 

These are the voyages of the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since.

== Origins of the theory ==

Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;&lt;ref name=Einstein1923/&gt;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.''

The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. 

Anco and Maghadam found that &lt;math&gt;SO(4)&lt;/math&gt; breaks to &lt;math&gt;\mathbb{S}^3&lt;/math&gt; (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to &lt;math&gt;H^3&lt;/math&gt; (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that &lt;math&gt;\mathbb{R}^4&lt;/math&gt; is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn|
...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}}

The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.&lt;ref&gt;{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}&lt;/ref&gt;}}

== Boundaries ==

&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the 
planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions?

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force.

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.&lt;ref&gt;{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}&lt;blockquote&gt;
:We shall not cease from exploration
:And the end of all our exploring
:Will be to arrive where we started
:And know the place for the first time.
:Through the unknown, remembered gate
:When the last of earth left to discover
:Is that which was the beginning;
:At the source of the longest river
:The voice of the hidden waterfall
:And the children in the apple-tree
:Not known, because not looked for
:But heard, half-heard, in the stillness
:Between two waves of the sea.
&lt;/blockquote&gt;&lt;/ref&gt;

Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity &lt;math&gt;c&lt;/math&gt;, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. 

I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== ... ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|name=radially equilateral}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}})
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Sfn|Mamone, Pileio &amp; Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

==Notes==
{{Regular convex 4-polytopes Notelist|wiki=W:}}

==Citations==
{{Regular convex 4-polytopes Reflist|wiki=W:}}

==References==
{{Refbegin}}
* {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}}
* {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}}

=== [[Polyscheme|Polyschemes]] ===
{{Regular convex 4-polytopes Refs|wiki=W:}}
{{Refend}}</text>
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  <page>
    <title>WikiJournal of Science/MALT1</title>
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        <username>OhanaUnited</username>
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      <text bytes="43175" sha1="o33607numh2jwwbeg0lsesnfm2q5r0g" xml:space="preserve">{{Article info
| first1 = Jens
| last1 = Staal
| orcid1 = 0000-0003-2664-3357
| affiliation1 = Ghent University
| correspondence1 = jens.staal@irc.vib-ugent.be
| journal = WikiJournal of Science
| w1 = MALT1
| from w1 = true
| submitted = 6 Feb 2023
| abstract = '''Mucosa-associated lymphoid tissue lymphoma translocation protein 1''' is a [[w:protein|protein]] that in humans is encoded by the ''MALT1'' [[w:gene|gene]].&lt;ref name=&quot;pmid10339464&quot;&gt;{{cite journal | authors = Dierlamm J, Baens M, Wlodarska I, Stefanova-Ouzounova M, Hernandez JM, Hossfeld DK, De Wolf-Peeters C, Hagemeijer A, Van den Berghe H, Marynen P | display-authors = 6 | title = The apoptosis inhibitor gene API2 and a novel 18q gene, MLT, are recurrently rearranged in the t(11;18)(q21;q21) associated with mucosa-associated lymphoid tissue lymphomas | journal = Blood | volume = 93 | issue = 11 | pages = 3601–9 | date = June 1999 | pmid = 10339464 | doi =  10.1182/blood.V93.11.3601}}&lt;/ref&gt;&lt;ref name=&quot;pmid10406266&quot;&gt;{{cite journal | authors = Hosaka S, Akamatsu T, Nakamura S, Kaneko T, Kitano K, Kiyosawa K, Ota H, Hosaka N, Miyabayashi H, Katsuyama T | display-authors = 6 | title = Mucosa-associated lymphoid tissue (MALT) lymphoma of the rectum with chromosomal translocation of the t(11;18)(q21;q21) and an additional aberration of trisomy 3 | journal = Am J Gastroenterol | volume = 94 | issue = 7 | pages = 1951–4 | date = July 1999 | doi = 10.1111/j.1572-0241.1999.01237.x | pmid = 10406266 | s2cid = 188457 }}&lt;/ref&gt;&lt;ref name=&quot;pmid10523859&quot;&gt;{{cite journal | authors = Akagi T, Motegi M, Tamura A, Suzuki R, Hosokawa Y, Suzuki H, Ota H, Nakamura S, Morishima Y, Taniwaki M, Seto M | display-authors = 6 | title = A novel gene, MALT1 at 18q21, is involved in t(11;18) (q21;q21) found in low-grade B-cell lymphoma of mucosa-associated lymphoid tissue | journal = Oncogene | volume = 18 | issue = 42 | pages = 5785–94 | date = November 1999 | pmid = 10523859 | doi = 10.1038/sj.onc.1203018 | doi-access = free }}&lt;/ref&gt; It's the human [[w:paracaspase|paracaspase]].
}}

== Function ==

[[File:Protein_MALT1_PDB_2g7r.png|thumb|Structure of the MALT1 protein ({{PDB|2G7R}}) {{attrib|[[User:Emw|Emw]]|[https://creativecommons.org/licenses/by-sa/3.0/deed.en CC-BY-SA 3.0]}}]]

Genetic ablation of the [[w:paracaspase|paracaspase]] gene ''Malt1'' in mice and biochemical studies have shown that MALT1 is a crucial protein for T and B [[w:lymphocytes|lymphocytes]] activation. It has an important role in the activation of the transcription factor [[w:NF-κB|NF-κB]], in the production of [[w:interleukin-2|interleukin-2]] (IL-2) and in T and B lymphocytes proliferation&lt;ref&gt;{{cite journal |authors=Ruefli-Brasse AA, French DM, Dixit VM |title=Regulation of NF-kappaB-dependent lymphocyte activation and development by paracaspase |journal=Science |volume=302|issue=5650 |pages=1581–4 |year=2003 |pmid=14576442|doi=10.1126/science.1090769|bibcode=2003Sci...302.1581R |s2cid=19381027 |url=https://semanticscholar.org/paper/74dcf0a2b3df77d05e12ad299887d384b389a179 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal |authors=Ruland J, Duncan GS, Wakeham A, Mak TW |title=Differential requirement for Malt1 in T and B cell antigen receptor signaling |journal=Immunity |volume=19 |issue=5 |pages=749–58 |year=2003 |pmid=14614861 |doi=10.1016/S1074-7613(03)00293-0|doi-access=free }}&lt;/ref&gt;  MALT1 plays a critical role in the adaptive immunity also in humans, and patients deficient for MALT1 show [[w:Combined immunodeficiencies|combined immunodeficiency]] (CID), failure to thrive, [[w:Osteoporosis|osteoporosis]] and [[w:Atopic dermatitis|atopic dermatitis]]. &lt;ref&gt;{{Cite journal|last=McKinnon|first=Margaret L.|last2=Rozmus|first2=Jacob|last3=Fung|first3=Shan-Yu|last4=Hirschfeld|first4=Aaron F.|last5=Del Bel|first5=Kate L.|last6=Thomas|first6=Leah|last7=Marr|first7=Nico|last8=Martin|first8=Spencer D.|last9=Marwaha|first9=Ashish K.|date=2014-05|title=Combined immunodeficiency associated with homozygous MALT1 mutations|url=https://linkinghub.elsevier.com/retrieve/pii/S0091674913017065|journal=Journal of Allergy and Clinical Immunology|language=en|volume=133|issue=5|pages=1458–1462.e7|doi=10.1016/j.jaci.2013.10.045}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Jabara|first=Haifa H.|last2=Ohsumi|first2=Toshiro|last3=Chou|first3=Janet|last4=Massaad|first4=Michel J.|last5=Benson|first5=Halli|last6=Megarbane|first6=Andre|last7=Chouery|first7=Eliane|last8=Mikhael|first8=Raymond|last9=Gorka|first9=Oliver|date=2013-07|title=A homozygous mucosa-associated lymphoid tissue 1 (MALT1) mutation in a family with combined immunodeficiency|url=https://linkinghub.elsevier.com/retrieve/pii/S0091674913006945|journal=Journal of Allergy and Clinical Immunology|language=en|volume=132|issue=1|pages=151–158|doi=10.1016/j.jaci.2013.04.047|pmc=PMC3700575|pmid=23727036}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Punwani|first=Divya|last2=Wang|first2=Haopeng|last3=Chan|first3=Alice Y.|last4=Cowan|first4=Morton J.|last5=Mallott|first5=Jacob|last6=Sunderam|first6=Uma|last7=Mollenauer|first7=Marianne|last8=Srinivasan|first8=Rajgopal|last9=Brenner|first9=Steven E.|date=2015-02|title=Combined Immunodeficiency Due to MALT1 Mutations, Treated by Hematopoietic Cell Transplantation|url=http://link.springer.com/10.1007/s10875-014-0125-1|journal=Journal of Clinical Immunology|language=en|volume=35|issue=2|pages=135–146|doi=10.1007/s10875-014-0125-1|issn=0271-9142|pmc=PMC4352191|pmid=25627829}}&lt;/ref&gt; The osteoporosis and atopic dermatitis phenotypes have been replicated in ''Malt1'' knock out mice.&lt;ref&gt;{{Cite journal|last=Demeyer|first=Annelies|last2=Van Nuffel|first2=Elien|last3=Baudelet|first3=Griet|last4=Driege|first4=Yasmine|last5=Kreike|first5=Marja|last6=Muyllaert|first6=David|last7=Staal|first7=Jens|last8=Beyaert|first8=Rudi|date=2019-10-01|title=MALT1-Deficient Mice Develop Atopic-Like Dermatitis Upon Aging|url=https://www.frontiersin.org/article/10.3389/fimmu.2019.02330/full|journal=Frontiers in Immunology|volume=10|doi=10.3389/fimmu.2019.02330|issn=1664-3224|pmc=PMC6779721|pmid=31632405}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Gilis|first=Elisabeth|last2=Gaublomme|first2=Djoere|last3=Staal|first3=Jens|last4=Venken|first4=Koen|last5=Dhaenens|first5=Maarten|last6=Lambrecht|first6=Stijn|last7=Coudenys|first7=Julie|last8=Decruy|first8=Tine|last9=Schryvers|first9=Nadia|date=2019-12|title=Deletion of Mucosa‐Associated Lymphoid Tissue Lymphoma Translocation Protein 1 in Mouse T Cells Protects Against Development of Autoimmune Arthritis but Leads to Spontaneous Osteoporosis|url=https://acrjournals.onlinelibrary.wiley.com/doi/10.1002/art.41029|journal=Arthritis &amp; Rheumatology|language=en|volume=71|issue=12|pages=2005–2015|doi=10.1002/art.41029|issn=2326-5191}}&lt;/ref&gt; Conditional ''Malt1'' mouse models show that the osteoporosis phenotype is dependent on MALT1 in T cells. On the other hand, it is still unknown what cell type is responsible for the atopic dermatitis phenotype (T cells and keratinocytes are excluded).

In addition, a role for MALT1 has been shown in the innate immune response mediated by the zymosan receptor  [[w:Dectin-1|Dectin-1]] in [[w:macrophages|macrophages]] and [[w:dendritic cells|dendritic cells]], and in response to the stimulation of certain [[w:G protein-coupled receptors|G protein-coupled receptors]].&lt;ref&gt;{{cite journal |authors=Wegener E, Krappmann D |title=CARD-Bcl10-Malt1 signalosomes: missing link to NF-kappaB |journal=Sci STKE |volume= 2007|year=2007 |pmid=17473310 |issue=384 |pages=pe21 |doi=10.1126/stke.3842007pe21|s2cid=86150342 }}&lt;/ref&gt; At this moment, most functional analyses of MALT1 have been performed in lymphocytes (T cells and B cell lymphomas). The role of MALT1 in other cell types and signaling pathways are much less understood.

Sequence analysis show that MALT1 has an [[w:N-terminal|N-terminal]] death domain, two central immunoglobulin-like domains involved in the binding to the B-cell lymphoma 10 ([[w:BCL10|BCL10]]) protein and a caspase-like (paracaspase) domain. Two alternatively spliced transcript variants encoding different isoforms have been described for this gene.&lt;ref&gt;{{cite web|url=https://www.ncbi.nlm.nih.gov/sites/entrez?Db=gene&amp;Cmd=ShowDetailView&amp;TermToSearch=10892|title=Entrez Gene: MALT1 mucosa associated lymphoid tissue lymphoma translocation gene 1}}&lt;/ref&gt; The death domain and immunoglobulin-like domains participate in binding to [[w:BCL10|BCL10]]. Activation of MALT1 downstream NF-κB signaling and protease activity occurs when BCL10/MALT1 gets recruited to an activated [[w:CARD-CC family|CARD-CC]] family protein ([[w:CARD9|CARD9]], [[w:CARD10|-10]], [[w:CARD11|-11]] or [[w:CARD14|-14]]) in a so-called '''CBM''' (CARD-CC/BCL10/MALT1) signaling complex.

MALT1, being a [[w:Paracaspase|paracaspase]], has been shown to have [[w:proteolytic|proteolytic]] activity through its caspase-like domain in T [[w:lymphocytes|lymphocytes]] and other cell types. [[w:Cysteine|Cysteine]] 464 and [[w:histidine|histidine]] 415 are crucial for this activity. Like metacaspases, the paracaspase cleaves substrates after an [[w:arginine|arginine]] residue. To date, several MALT1 protease substrates have been described (see below). The role of specific substrate cleavage events are unclear, but the MALT1 protease activity plays a critical role in the regulation of the immune system.

&quot;Protease dead&quot; knock-in mice where the MALT1 catalytic cysteine has been mutated to an alanine show lethal autoimmunity, &lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Gewies|first=Andreas|last2=Gorka|first2=Oliver|last3=Bergmann|first3=Hanna|last4=Pechloff|first4=Konstanze|last5=Petermann|first5=Franziska|last6=Jeltsch|first6=Katharina M.|last7=Rudelius|first7=Martina|last8=Kriegsmann|first8=Mark|last9=Weichert|first9=Wilko|date=2014-11|title=Uncoupling Malt1 Threshold Function from Paracaspase Activity Results in Destructive Autoimmune Inflammation|url=https://linkinghub.elsevier.com/retrieve/pii/S2211124714009139|journal=Cell Reports|language=en|volume=9|issue=4|pages=1292–1305|doi=10.1016/j.celrep.2014.10.044}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Yu|first=Jong W.|last2=Hoffman|first2=Sandy|last3=Beal|first3=Allison M.|last4=Dykon|first4=Angela|last5=Ringenberg|first5=Michael A.|last6=Hughes|first6=Anna C.|last7=Dare|first7=Lauren|last8=Anderson|first8=Amber D.|last9=Finger|first9=Joshua|date=2015-05-12|editor-last=Krieg|editor-first=Andreas|title=MALT1 Protease Activity Is Required for Innate and Adaptive Immune Responses|url=https://dx.plos.org/10.1371/journal.pone.0127083|journal=PLOS ONE|language=en|volume=10|issue=5|pages=e0127083|doi=10.1371/journal.pone.0127083|issn=1932-6203|pmc=PMC4428694|pmid=25965667}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Jaworski|first=Maike|last2=Marsland|first2=Ben J|last3=Gehrig|first3=Jasmine|last4=Held|first4=Werner|last5=Favre|first5=Stéphanie|last6=Luther|first6=Sanjiv A|last7=Perroud|first7=Mai|last8=Golshayan|first8=Déla|last9=Gaide|first9=Olivier|date=2014-12|title=Malt1 protease inactivation efficiently dampens immune responses but causes spontaneous autoimmunity|url=https://www.embopress.org/doi/10.15252/embj.201488987|journal=The EMBO Journal|language=en|volume=33|issue=23|pages=2765–2781|doi=10.15252/embj.201488987|issn=0261-4189|pmc=PMC4282555|pmid=25319413}}&lt;/ref&gt; and conditional mutant mice have shown that this MALT1 protease-dependent immune regulation is taking place in T cells,&lt;ref&gt;{{Cite journal|last=Demeyer|first=Annelies|last2=Skordos|first2=Ioannis|last3=Driege|first3=Yasmine|last4=Kreike|first4=Marja|last5=Hochepied|first5=Tino|last6=Baens|first6=Mathijs|last7=Staal|first7=Jens|last8=Beyaert|first8=Rudi|date=2019-08-14|title=MALT1 Proteolytic Activity Suppresses Autoimmunity in a T Cell Intrinsic Manner|url=https://www.frontiersin.org/article/10.3389/fimmu.2019.01898/full|journal=Frontiers in Immunology|volume=10|doi=10.3389/fimmu.2019.01898|issn=1664-3224|pmc=PMC6702287|pmid=31474984}}&lt;/ref&gt; and especially in [[w:Regulatory T cell|regulatory T cell]]s.&lt;ref&gt;{{Cite journal|last=Rosenbaum|first=Marc|last2=Gewies|first2=Andreas|last3=Pechloff|first3=Konstanze|last4=Heuser|first4=Christoph|last5=Engleitner|first5=Thomas|last6=Gehring|first6=Torben|last7=Hartjes|first7=Lara|last8=Krebs|first8=Sabrina|last9=Krappmann|first9=Daniel|date=2019-05-28|title=Bcl10-controlled Malt1 paracaspase activity is key for the immune suppressive function of regulatory T cells|url=https://www.nature.com/articles/s41467-019-10203-2|journal=Nature Communications|language=en|volume=10|issue=1|doi=10.1038/s41467-019-10203-2|issn=2041-1723|pmc=PMC6538646|pmid=31138793}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Rosenbaum|first=Marc|last2=Schnalzger|first2=Theresa|last3=Engleitner|first3=Thomas|last4=Weiß|first4=Christin|last5=Mishra|first5=Ritu|last6=Mibus|first6=Cora|last7=Mitterer|first7=Theresa|last8=Rad|first8=Roland|last9=Ruland|first9=Jürgen|date=2022-01|title=MALT1 protease function in regulatory T cells induces MYC activity to promote mitochondrial function and cellular expansion|url=https://onlinelibrary.wiley.com/doi/10.1002/eji.202149355|journal=European Journal of Immunology|language=en|volume=52|issue=1|pages=85–95|doi=10.1002/eji.202149355|issn=0014-2980}}&lt;/ref&gt; The lethal autoimmunity is driven by IFNγ-producing T cells.&lt;ref name=&quot;:0&quot; /&gt; Until now, no uncleavable knock-in substrate mutant mouse has replicated this phenotype, which means that it is unclear how MALT1 protease activity is regulating immune homeostasis.&lt;ref&gt;{{Cite journal|last=Moud|first=Bahareh Nemati|last2=Ober|first2=Franziska|last3=O’Neill|first3=Thomas J.|last4=Krappmann|first4=Daniel|date=2024-05-28|title=MALT1 substrate cleavage: what is it good for?|url=https://www.frontiersin.org/articles/10.3389/fimmu.2024.1412347/full|journal=Frontiers in Immunology|volume=15|doi=10.3389/fimmu.2024.1412347|issn=1664-3224|pmc=PMC11165066|pmid=38863711}}&lt;/ref&gt; Interestingly, it seems like MALT1 protease inhibition in adult animals leads to a less severe autoimmunity,&lt;ref&gt;{{Cite journal|last=Martin|first=Kea|last2=Junker|first2=Ursula|last3=Tritto|first3=Elaine|last4=Sutter|first4=Esther|last5=Rubic-Schneider|first5=Tina|last6=Morgan|first6=Hannah|last7=Niwa|first7=Satoru|last8=Li|first8=Jianping|last9=Schlapbach|first9=Achim|date=2020-04-30|title=Pharmacological Inhibition of MALT1 Protease Leads to a Progressive IPEX-Like Pathology|url=https://www.frontiersin.org/article/10.3389/fimmu.2020.00745/full|journal=Frontiers in Immunology|volume=11|doi=10.3389/fimmu.2020.00745|issn=1664-3224|pmc=PMC7203682|pmid=32425939}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Demeyer|first=Annelies|last2=Driege|first2=Yasmine|last3=Skordos|first3=Ioannis|last4=Coudenys|first4=Julie|last5=Lemeire|first5=Kelly|last6=Elewaut|first6=Dirk|last7=Staal|first7=Jens|last8=Beyaert|first8=Rudi|date=2020-10|title=Long-Term MALT1 Inhibition in Adult Mice Without Severe Systemic Autoimmunity|url=https://linkinghub.elsevier.com/retrieve/pii/S2589004220307495|journal=iScience|language=en|volume=23|issue=10|pages=101557|doi=10.1016/j.isci.2020.101557|pmc=PMC7522757|pmid=33083726}}&lt;/ref&gt; which might indicate that thymic regulatory T cells, which develop at an early age, are important for the suppression of autoimmunity. This is also encouraging for the potential therapeutic use of MALT1 protease inhibitors: 

By targeting MALT1 proteolytic activity, it might be possible to develop new drugs that might be useful for the treatment of certain [[w:lymphomas|lymphomas]], solid tumors or [[w:autoimmune|autoimmune]] disorders. Not only by affecting the cancer cells themselves, but also to enhance immune check point inhibition by suppressing regulatory T cells, enhance cancer cell killing by promoting IFNγ-producing T cells, and possibly also influence the polarization of tumor-associated macrophages towards a more pro-inflammatory phenotype.&lt;ref&gt;{{Cite journal| publisher=bioRxiv| doi=10.1101/2024.09.26.614808| last1=Azambuja| first1=Juliana Hofstätter| last2=Yerneni| first2=Saigopalakrishna S.| last3=Maurer| first3=Lisa M.| last4=Crentsil| first4=Hannah E.| last5=Debom| first5=Gabriela N.| last6=Klei| first6=Linda| last7=Smyers| first7=Mei| last8=Sneiderman| first8=Chaim T.| last9=Schwab| first9=Kristina E.| last10=Acharya| first10=Rajesh| last11=Wu| first11=Yijen Lin| last12=Ekambaram| first12=Prasanna| last13=Hu| first13=Dong| last14=Gough| first14=Pete J.| last15=Bertin| first15=John| last16=Melnick| first16=Ari| last17=Kohanbash| first17=Gary| last18=Bao| first18=Riyue| last19=Lucas| first19=Peter C.| last20=McAllister-Lucas| first20=Linda M.| title=MALT1 protease inhibition restrains glioblastoma progression by reversing tumor-associated macrophage-dependent immunosuppression| access-date=2024-10-23| date=2024-09-27| url=https://www.biorxiv.org/content/10.1101/2024.09.26.614808v1}}&lt;/ref&gt;

== Interactions ==

MALT1 has been shown to [[w:Protein-protein interaction|interact]] with [[w:BCL10|BCL10]],&lt;ref name=pmid11090634&gt;{{cite journal | authors = Uren AG, O'Rourke K, Aravind LA, Pisabarro MT, Seshagiri S, Koonin EV, Dixit VM | title = Identification of paracaspases and metacaspases: two ancient families of caspase-like proteins, one of which plays a key role in MALT lymphoma | journal = Mol. Cell | volume = 6 | issue = 4 | pages = 961–7 | date = October 2000 | pmid = 11090634 | doi = 10.1016/S1097-2765(05)00086-9 | doi-access = free }}&lt;/ref&gt; [[w:TRAF6|TRAF6]] and [[w:Sequestosome 1|SQSTM1/p62]]. The scaffold activity of MALT1 binding TRAF6 is critical for [[w:NF-κB|NF-κB]] activation downstream of CBM complex formation. In addition to [[w:NF-κB|NF-κB]] activation, MALT1 is also important for [[w:MTOR|mTOR]] activation in T cells and keratinocytes.&lt;ref&gt;{{Cite journal|last=Hamilton|first=Kristia S.|last2=Phong|first2=Binh|last3=Corey|first3=Catherine|last4=Cheng|first4=Jing|last5=Gorentla|first5=Balachandra|last6=Zhong|first6=Xiaoping|last7=Shiva|first7=Sruti|last8=Kane|first8=Lawrence P.|date=2014-06-10|title=T Cell Receptor–Dependent Activation of mTOR Signaling in T Cells Is Mediated by Carma1 and MALT1, But Not Bcl10|url=https://www.science.org/doi/10.1126/scisignal.2005169|journal=Science Signaling|language=en|volume=7|issue=329|doi=10.1126/scisignal.2005169|issn=1945-0877|pmc=PMC4405131|pmid=24917592}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=O'Sullivan|first=Paul A.|last2=Aidarova|first2=Aigerim|last3=Afonina|first3=Inna S.|last4=Manils|first4=Joan|last5=Thurston|first5=Teresa L. M.|last6=Instrell|first6=Rachael|last7=Howell|first7=Michael|last8=Boeing|first8=Stefan|last9=Ranawana|first9=Sashini|date=2024-09-18|title=CARD14 signalosome formation is associated with its endosomal relocation and mTORC1-induced keratinocyte proliferation|url=https://portlandpress.com/biochemj/article/481/18/1143/234816/CARD14-signalosome-formation-is-associated-with|journal=Biochemical Journal|language=en|volume=481|issue=18|pages=1143–1171|doi=10.1042/BCJ20240058|issn=0264-6021}}&lt;/ref&gt;

== Protease substrates ==

MALT1 (PCASP1) is part of the [[w:paracaspase|paracaspase]] family and shows proteolytic activity. Since many of the substrates are involved in regulation of inflammatory responses, the protease activity of MALT1 has emerged as an interesting therapeutic target. The general pattern of MALT1 substrate specificity is a hydrophobic amino acid at P4 and small uncharged residues surrounding (P2 and P1') the substrate (P1) arginine. There are however exceptions to this rule. For example, the tetrapeptide GASR is a poor substrate,&lt;ref&gt;{{Cite journal|last=Hachmann|first=Janna|last2=Snipas|first2=Scott J.|last3=van Raam|first3=Bram J.|last4=Cancino|first4=Erik M.|last5=Houlihan|first5=Emily J.|last6=Poreba|first6=Marcin|last7=Kasperkiewicz|first7=Paulina|last8=Drag|first8=Marcin|last9=Salvesen|first9=Guy S.|date=2012-04-01|title=Mechanism and specificity of the human paracaspase MALT1|url=https://portlandpress.com/biochemj/article/443/1/287/80330/Mechanism-and-specificity-of-the-human-paracaspase|journal=Biochemical Journal|language=en|volume=443|issue=1|pages=287–295|doi=10.1042/BJ20120035|issn=0264-6021|pmc=PMC3304489|pmid=22309193}}&lt;/ref&gt; and for A20 it has been shown that the P5 leucine (LGASR) is required for A20 cleavage.&lt;ref&gt;{{Cite journal|last=Hulpiau|first=Paco|last2=Driege|first2=Yasmine|last3=Staal|first3=Jens|last4=Beyaert|first4=Rudi|date=2016-03|title=MALT1 is not alone after all: identification of novel paracaspases|url=http://link.springer.com/10.1007/s00018-015-2041-9|journal=Cellular and Molecular Life Sciences|language=en|volume=73|issue=5|pages=1103–1116|doi=10.1007/s00018-015-2041-9|issn=1420-682X|pmc=PMC11108557|pmid=26377317}}&lt;/ref&gt; Currently known protease substrates are (in order of reported discovery):
{| class=&quot;wikitable&quot;
|+ MALT1 protease substrates
|-
! Substrate !! Reference !! Cleavage sequence
|-
| A20 ([[w:TNFAIP3|TNFAIP3]])|| &lt;ref name=&quot;Coornaert_2008&quot;&gt;{{cite journal|authors=Coornaert B, Baens M, Heyninck K, Bekaert T, Haegman M, Staal J, Sun L, Chen ZJ, Marynen P, Beyaert R|year=2008|title=T cell antigen receptor stimulation induces MALT1 paracaspase-mediated cleavage of the NF-kappaB inhibitor A20.|journal=Nature Immunology|volume=9|issue=3|pages=263–71|doi=10.1038/ni1561|pmid=18223652|s2cid=29300246}}&lt;/ref&gt;|| LGASR/G
|-
|[[w:BCL10|BCL10]]|| &lt;ref name=&quot;Rebeaud_2008&quot;&gt;{{cite journal|authors=Rebeaud F, Hailfinger S, Posevitz-Fejfar A, Tapernoux M, Moser R, Rueda D, Gaide O, Guzzardi M, Iancu EM, Rufer N, Fasel N, Thome M|year=2008|title=The proteolytic activity of the paracaspase MALT1 is key in T cell activation.|journal=Nature Immunology|volume=9|issue=3|pages=272–81|doi=10.1038/ni1568|pmid=18264101|s2cid=205361198}}&lt;/ref&gt;|| LRSR/T
|-
| [[w:Cylindromatosis|CYLD]] || &lt;ref&gt;{{cite journal |authors=Staal J, Driege Y, Bekaert T, Demeyer A, Muyllaert D, Van Damme P, Gevaert K, Beyaert R |title=T-cell receptor-induced JNK activation requires proteolytic inactivation of CYLD by MALT1. |journal=EMBO J. |volume=30 |year=2011 |pmid=21448133 |doi=10.1038/emboj.2011.85 |pages=1742–52 |issue=4 |pmc=3101995}}&lt;/ref&gt; || FMSR/G
|-
| [[w:RELB|RELB]] || &lt;ref&gt;{{cite journal |authors=Hailfinger S, Nogai H, Pelzer C, Jaworski M, Cabalzar K, Charton JE, Guzzardi M, Décaillet C, Grau M, Dörken B, Lenz P, Lenz G, Thome M |title=Malt1-dependent RelB cleavage promotes canonical NF-kappaB activation in lymphocytes and lymphoma cell lines. |journal=Proc Natl Acad Sci U S A  |volume=108 |year=2011 |pmid=21873235 |doi=10.1073/pnas.1105020108 |pages=14596–601 |issue=35 |pmc=3167514|bibcode=2011PNAS..10814596H |doi-access=free }}&lt;/ref&gt; || LVSR/G
|-
|regnase-1/MCPIP1 ([[w:ZC3H12A|ZC3H12A]]) ||  &lt;ref&gt;{{cite journal |authors=Uehata T, Iwasaki H, Vandenbon A, Matsushita K, Hernandez-Cuellar E, Kuniyoshi K, Satoh T, Mino T, Suzuki Y, Standley DM, Tsujimura T, Rakugi H, Isaka Y, Takeuchi O, Akira S |title=Malt1-induced cleavage of regnase-1 in CD4(+) helper T cells regulates immune activation. |journal=Cell |volume= 153|year=2013 |pmid=    23706741 |doi=10.1016/j.cell.2013.04.034 |pages=1036–49 |issue=5 |doi-access=free }}&lt;/ref&gt; || LVPR/G
|-
| Roquin-1([[w:RC3H1|RC3H1]]) || &lt;ref name=&quot;Jeltsch_2014&quot;&gt; {{cite journal |authors=Jeltsch KM, Hu D, Brenner S, Zöller J, Heinz GA, Nagel D, Vogel KU, Rehage N, Warth SC, Edelmann SL, Gloury R, Martin N, Lohs C, Lech M, Stehklein JE, Geerlof A, Kremmer E, Weber A, Anders HJ, Schmitz I, Schmidt-Supprian M, Fu M, Holtmann H, Krappmann D, Ruland J, Kallies A, Heikenwalder M, Heissmeyer V | display-authors = 6 |title=Cleavage of roquin and regnase-1 by the paracaspase MALT1 releases their cooperatively repressed targets to promote T(H)17 differentiation. |journal=Nat Immunol  |volume=15 |year=2014 |pmid=    25282160 |doi=10.1038/ni.3008 |pages=1079–89 |issue=11 | s2cid = 8140814 }}&lt;/ref&gt; || LIPR/G
|-
| Roquin-2([[w:RC3H2|RC3H2]])|| &lt;ref name=&quot;Jeltsch_2014&quot;/&gt; || LISR/S
|-
| MALT1 auto-proteolysis || &lt;ref&gt;{{cite journal |authors=Baens M, Bonsignore L, Somers R, Vanderheydt C, Weeks SD, Gunnarsson J, Nilsson E, Roth RG, Thome M, Marynen P |title=MALT1 auto-proteolysis is essential for NF-κB-dependent gene transcription in activated lymphocytes. |journal=PLOS ONE |volume=9 |issue=8 |year=2014 |pmid=25105596 |doi=10.1371/journal.pone.0103774 |pages=e103774 |pmc=4126661|bibcode=2014PLoSO...9j3774B |doi-access=free }}&lt;/ref&gt; || LCCR/A
|-
| MALT1 auto-proteolysis || &lt;ref&gt;{{cite journal | authors = Ginster S, Bardet M, Unterreiner A, Malinverni C, Renner F, Lam S, Freuler F, Gerrits B, Voshol J, Calzascia T, Régnier CH, Renatus M, Nikolay R, Israël L, Bornancin F | display-authors = 6 | title = Two Antagonistic MALT1 Auto-Cleavage Mechanisms Reveal a Role for TRAF6 to Unleash MALT1 Activation | journal = PLOS ONE | volume = 12 | issue = 1 | pages = e0169026 | date = 2017 | pmid = 28052131 | doi = 10.1371/journal.pone.0169026 | pmc = 5214165 | bibcode = 2017PLoSO..1269026G | doi-access = free }}&lt;/ref&gt; || HCSR/T
|-
| HOIL1 ([[w:RBCK1|RBCK1]]) || &lt;ref&gt;{{cite journal |authors=Klein T, Fung SY, Renner F, Blank MA, Dufour A, Kang S, Bolger-Munro M, Scurll JM, Priatel JJ, Schweigler P, Melkko S, Gold MR, Viner RI, Régnier CH, Turvey SE, Overall CM |title=The paracaspase MALT1 cleaves HOIL1 reducing linear ubiquitination by LUBAC to dampen lymphocyte NF-κB signalling. |journal=Nat. Commun. |volume= 6|year=2015 |pmid=26525107 |doi=10.1038/ncomms9777 |pages= 8777 |pmc=4659944|bibcode=2015NatCo...6.8777K }}&lt;/ref&gt;&lt;ref&gt;{{cite journal |authors=Elton L, Carpentier I, Staal J, Driege Y, Haegman M, Beyaert R |title=MALT1 cleaves the E3 ubiquitin ligase HOIL-1 in activated T cells, generating a dominant negative inhibitor of LUBAC-induced NF-κB signaling. |journal=FEBS J. |volume= 283|issue=3 |year=2015 |pmid=26573773 |doi=10.1111/febs.13597 |pages= 403–12|doi-access=free }}&lt;/ref&gt;&lt;ref&gt;{{cite journal | authors = Douanne T, Gavard J, Bidère N | title = The paracaspase MALT1 cleaves the LUBAC subunit HOIL1 during antigen receptor signaling | journal = Journal of Cell Science | volume = 129 | issue = 9 | pages = 1775–80 | date = May 2016 | pmid = 27006117 | doi = 10.1242/jcs.185025 | s2cid = 31415485 | url = http://www.hal.inserm.fr/inserm-01311283/file/ArticleEqu15_2016.pdf | doi-access = free }}&lt;/ref&gt; || LQPR/G
|-
| [https://www.genecards.org/cgi-bin/carddisp.pl?gene=N4BP1&amp;keywords=n4bp1 N4BP1] || &lt;ref&gt;{{cite journal | authors = Yamasoba D, Sato K, Ichinose T, Imamura T, Koepke L, Joas S, Reith E, Hotter D, Misawa N, Akaki K, Uehata T, Mino T, Miyamoto S, Noda T, Yamashita A, Standley DM, Kirchhoff F, Sauter D, Koyanagi Y, Takeuchi O | display-authors = 6 | title = N4BP1 restricts HIV-1 and its inactivation by MALT1 promotes viral reactivation | journal = Nature Microbiology | volume = 4 | issue = 9 | pages = 1532–1544 | date = May 2019 | pmid = 31133753 | doi = 10.1038/s41564-019-0460-3 | hdl = 2433/244207 | s2cid = 167207661 | hdl-access = free }}&lt;/ref&gt; || FVSR/G
|-
|[[w:CARD10|CARD10]]|| &lt;ref name=&quot;Israel2021&quot;&gt;{{cite journal| author=Israël L, Glück A, Berger M, Coral M, Ceci M, Unterreiner A | display-authors=etal| title=CARD10 cleavage by MALT1 restricts lung carcinoma growth in vivo. | journal=Oncogenesis | year= 2021 | volume= 10 | issue= 4 | pages= 32 | pmid=33824280 | doi=10.1038/s41389-021-00321-2 | pmc=8024357 }}&lt;/ref&gt; || LRCR/G
|-
| [https://www.genecards.org/cgi-bin/carddisp.pl?gene=ZC3H12D ZC3H12D] || &lt;ref name=&quot;Bell_2022&quot;&gt; {{Cite journal| doi = 10.1016/j.csbj.2022.08.021| issn = 2001-0370| last1 = Bell| first1 = Peter A.| last2 = Scheuermann| first2 = Sophia| last3 = Renner| first3 = Florian| last4 = Pan| first4 = Christina L.| last5 = Lu| first5 = Henry Y.| last6 = Turvey| first6 = Stuart E.| last7 = Bornancin| first7 = Frédéric| last8 = Régnier| first8 = Catherine H.| last9 = Overall| first9 = Christopher M.| title = Integrating knowledge of protein sequence with protein function for the prediction and validation of new MALT1 substrates| journal = Computational and Structural Biotechnology Journal| date = 2022-08-19| volume = 20| pages = 4717–4732| pmid = 36147669| pmc = 9463181}} &lt;/ref&gt; || LVPR/G
|-
|[[w:ZC3H12B|ZC3H12B]]|| &lt;ref name=&quot;Bell_2022&quot;/&gt; || LVPR/G
|-
|[[w:MAP3K7IP3|TAB3]]|| &lt;ref name=&quot;Bell_2022&quot;/&gt; || LQSR/G
|-
| [[w:Caspase_10|CASP10]]|| &lt;ref name=&quot;Bell_2022&quot;/&gt; || LVSR/G
|-
| [https://www.genecards.org/cgi-bin/carddisp.pl?gene=CILK1&amp;keywords=cilk1 CILK1] || &lt;ref name=&quot;Bell_2022&quot;/&gt; || LISR/S
|-
| [https://www.genecards.org/cgi-bin/carddisp.pl?gene=ILDR2&amp;keywords=ildr2 ILDR2] || &lt;ref name=&quot;Bell_2022&quot;/&gt; || GASR/G LVSR/T GASR/G
|-
| [[w:TANK_(gene)|TANK]]|| &lt;ref name=&quot;Bell_2022&quot;/&gt; || HIPR/V
|}

Specifically by the oncogenic [[w:BIRC3|IAP2]]-MALT1 fusion:
* [[w:MAP3K14|NIK]]&lt;ref&gt;{{cite journal |authors=Rosebeck S, Madden L, Jin X, Gu S, Apel IJ, Appert A, Hamoudi RA, Noels H, Sagaert X, Van Loo P, Baens M, Du MQ, Lucas PC, McAllister-Lucas LM |title=Cleavage of NIK by the API2-MALT1 fusion oncoprotein leads to noncanonical NF-kappaB activation. |journal=Science  |volume=331 |year=2011 |pmid= 21273489 |doi=10.1126/science.1198946 |pages=468–72 |issue=6016 |pmc=3124150|bibcode=2011Sci...331..468R }}&lt;/ref&gt;
* [[w:LIMA1|LIMA1]]&lt;ref&gt;{{cite journal |authors=Nie Z, Du MQ, McAllister-Lucas LM, Lucas PC, Bailey NG, Hogaboam CM, Lim MS, Elenitoba-Johnson KS |title=Conversion of the LIMA1 tumour suppressor into an oncogenic LMO-like protein by API2-MALT1 in MALT lymphoma. |journal=Nat. Commun. |volume=6 |year=2015 |pmid= 25569716 |doi=10.1038/ncomms6908 |issue=5908 |pages=5908 |bibcode=2015NatCo...6.5908N |doi-access=free }}&lt;/ref&gt;

== Protease inhibitors ==
Since MALT1 protease activity is a promising therapeutic target, different screenings have been performed which have resulted in several types of protease inhibitors.&lt;ref&gt;{{cite journal | authors = Demeyer A, Staal J, Beyaert R | title = Targeting MALT1 Proteolytic Activity in Immunity, Inflammation and Disease: Good or Bad? | journal = Trends in Molecular Medicine | volume = 22 | issue = 2 | pages = 135–150 | date = February 2016 | pmid = 26787500 | doi = 10.1016/j.molmed.2015.12.004 }}&lt;/ref&gt; There is active competition between multiple pharma companies and independent research groups in drug development against the MALT1 protease activity.&lt;ref name=&quot;malt1patent&quot;&gt;{{cite journal| author=Hamp I, O'Neill TJ, Plettenburg O, Krappmann D| title=A patent review of MALT1 inhibitors (2013-present). | journal=Expert Opin Ther Pat | year= 2021 | volume=  31| issue=  12| pages= 1079–1096 | pmid=34214002 | doi=10.1080/13543776.2021.1951703 | pmc= | s2cid=235723498 | url=https://www.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?dbfrom=pubmed&amp;tool=sumsearch.org/cite&amp;retmode=ref&amp;cmd=prlinks&amp;id=34214002  }} &lt;/ref&gt;

==== Active site inhibitors ====
* Substrate [[w:peptide|peptide]]-based active-site inhibitor: Initially described with the [[w:metacaspase|metacaspase]] inhibitor VRPR-fmk.&lt;ref name=&quot;Rebeaud_2008&quot; /&gt; Others have developed peptide inhibitors based on the optimal peptide sequence (LVSR) or further chemical modifications. 
* [[w:Janssen Pharmaceutica|Janssen Pharmaceutica]] is currently performing a [[w:clinical trial|clinical trial]] with this class of inhibitors.&lt;ref&gt;Clinical trial number [https://www.clinicaltrials.gov/ct2/show/NCT03900598NCT03900598| NCT03900598] for &quot;A Study of JNJ-67856633 in Participants With Non-Hodgkin's Lymphoma (NHL) and Chronic Lymphocytic Leukemia (CLL)&quot; at [[w:ClinicalTrials.gov|ClinicalTrials.gov]]&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Philippar|first=Ulrike|last2=Lu|first2=Tianbao|last3=Vloemans|first3=Nele|last4=Bekkers|first4=Mariette|last5=Van Nuffel|first5=Luc|last6=Gaudiano|first6=Marcello|last7=Wnuk-Lipinska|first7=Katarzyna|last8=Van Der Leede|first8=Bas-jan|last9=Amssoms|first9=Katie|date=2020-08-15|title=Abstract 5690: Discovery of JNJ-67856633: A novel, first-in-class MALT1 protease inhibitor for the treatment of B cell lymphomas|url=https://aacrjournals.org/cancerres/article/80/16_Supplement/5690/644154/Abstract-5690-Discovery-of-JNJ-67856633-A-novel|journal=Cancer Research|language=en|volume=80|issue=16_Supplement|pages=5690–5690|doi=10.1158/1538-7445.AM2020-5690|issn=0008-5472}}&lt;/ref&gt;
* A [[w:molecular modeling|molecular modeling]] approach led to the development of the small molecule active site inhibitor MI-2.&lt;ref&gt;{{cite journal | authors = Fontan L, Yang C, Kabaleeswaran V, Volpon L, Osborne MJ, Beltran E, Garcia M, Cerchietti L, Shaknovich R, Yang SN, Fang F, Gascoyne RD, Martinez-Climent JA, Glickman JF, Borden K, Wu H, Melnick A | display-authors = 6 | title = MALT1 small molecule inhibitors specifically suppress ABC-DLBCL in vitro and in vivo | journal = Cancer Cell | volume = 22 | issue = 6 | pages = 812–24 | date = December 2012 | pmid = 23238016 | pmc = 3984478 | doi = 10.1016/j.ccr.2012.11.003 }}&lt;/ref&gt;

==== Allosteric inhibitors ====

*[[w:Phenothiazine|Phenothiazine]] compounds like [[w:mepazine|mepazine]] and [[w:chlorpromazine|chlorpromazine]] (which have been used clinically for neurological/psychological conditions) have been found to be [[w:allosteric|allosteric]] inhibitors of MALT1 protease activity.&lt;ref&gt;{{cite journal|authors=Nagel D, Spranger S, Vincendeau M, Grau M, Raffegerst S, Kloo B, Hlahla D, Neuenschwander M, Peter von Kries J, Hadian K, Dörken B, Lenz P, Lenz G, Schendel DJ, Krappmann D|display-authors=6|date=December 2012|title=Pharmacologic inhibition of MALT1 protease by phenothiazines as a therapeutic approach for the treatment of aggressive ABC-DLBCL|journal=Cancer Cell|volume=22|issue=6|pages=825–37|doi=10.1016/j.ccr.2012.11.002|pmid=23238017|doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|authors=Jacobs KA, André-Grégoire G, Maghe C, Li Y, Thys A, Harford-Wright E, Trillet K, Douanne T, Frénel JS, Bidère N, Gavard J|display-authors=6|date=2019-01-01|title=Control of the Endo-Lysosome Homeostasis by the Paracaspase MALT1 regulates Glioma Cell Survival|journal=bioRxiv|pages=582221|doi=10.1101/582221|doi-access=free}}&lt;/ref&gt;
* [[w:Biperiden|Biperiden]], like phenothiazines, act as a MALT1 protease inhibitor and show promising results against [[w:pancreatic cancer|pancreatic cancer]].&lt;ref&gt;{{Cite journal|last1=Konczalla|first1=Leonie|last2=Perez|first2=Daniel R.|last3=Wenzel|first3=Nadine|last4=Wolters-Eisfeld|first4=Gerrit|last5=Klemp|first5=Clarissa|last6=Lüddeke|first6=Johanna|last7=Wolski|first7=Annika|last8=Landschulze|first8=Dirk|last9=Meier|first9=Chris|date=2019-07-10|title=Biperiden and Mepazine effectively inhibit MALT1 activity and tumor growth in pancreatic cancer|journal=International Journal of Cancer|volume=146|issue=6|pages=1618–1630|doi=10.1002/ijc.32567|issn=1097-0215|pmid=31291468|doi-access=free|last10=Buchholz|first10=Anika|last11=Yao|first11=Dichao|last12=Hofmann|first12=Bianca T.|last13=Graß|first13=Julia K.|last14=Spriestersbach|first14=Sarah L.|last15=Grupp|first15=Katharina|last16=Schumacher|first16=Udo|last17=Betzel|first17=Christian|last18=Kapis|first18=Svetlana|last19=Nuguid|first19=Theresa|last20=Steinberg|first20=Pablo|last21=Püschel|first21=Klaus|last22=Sauter|first22=Guido|last23=Bockhorn|first23=Maximillian|last24=Uzunoglu|first24=Faik G.|last25=Izbicki|first25=Jakob R.|last26=Güngör|first26=Cenap|last27=El Gammal|first27=Alexander T.}}&lt;/ref&gt;
* [[w:Quinoline|Quinoline]] and thiazolopyridine allosteric MALT1 protease inhibitors have been demonstrated to work in mouse disease models.&lt;ref&gt;{{cite journal|authors=Scott DA, Hatcher JM, Liu H, Fu M, Du G, Fontán L, Us I, Casalena G, Qiao Q, Wu H, Melnick A, Gray NS|date=May 2019|title=Quinoline and thiazolopyridine allosteric inhibitors of MALT1|journal=Bioorganic &amp; Medicinal Chemistry Letters|volume=29|issue=14|pages=1694–1698|doi=10.1016/j.bmcl.2019.05.040|pmid=31129051|s2cid=167206456}}&lt;/ref&gt;
* [[w:Novartis|Novartis]] is developing [[w:pyrazolopyrimidine|pyrazolopyrimidine]] derivative MALT1 protease inhibitors.&lt;ref&gt;{{cite journal|authors=Bardet M, Unterreiner A, Malinverni C, Lafossas F, Vedrine C, Boesch D, Kolb Y, Kaiser D, Glück A, Schneider MA, Katopodis A, Renatus M, Simic O, Schlapbach A, Quancard J, Régnier CH, Bold G, Pissot-Soldermann C, Carballido JM, Kovarik J, Calzascia T, Bornancin F|display-authors=6|date=January 2018|title=The T-cell fingerprint of MALT1 paracaspase revealed by selective inhibition|journal=Immunology and Cell Biology|volume=96|issue=1|pages=81–99|doi=10.1111/imcb.1018|pmid=29359407|s2cid=36376441}}&lt;/ref&gt;&lt;ref&gt;{{cite journal|authors=Quancard J, Klein T, Fung SY, Renatus M, Hughes N, Israël L, Priatel JJ, Kang S, Blank MA, Viner RI, Blank J, Schlapbach A, Erbel P, Kizhakkedathu J, Villard F, Hersperger R, Turvey SE, Eder J, Bornancin F, Overall CM|display-authors=6|date=March 2019|title=An allosteric MALT1 inhibitor is a molecular corrector rescuing function in an immunodeficient patient|journal=Nature Chemical Biology|volume=15|issue=3|pages=304–313|doi=10.1038/s41589-018-0222-1|pmid=30692685|s2cid=59340695}}&lt;/ref&gt;
* [[w:AstraZeneca|AstraZeneca]] is developing MALT1 protease inhibitors.&lt;ref name=&quot;germinating&quot; /&gt;&lt;ref name=&quot;pmid34742013&quot;&gt;{{cite journal|authors=Schiesser S, Hajek P, Pople HE, Käck H, Öster L, Cox RJ|date=October 2021|title=Discovery and optimization of cyclohexane-1,4-diamines as allosteric MALT1 inhibitors|url=https://orca.cardiff.ac.uk/id/eprint/146131/1/PP.pdf|journal=European Journal of Medicinal Chemistry|volume=227|issue=|pages=113925|doi=10.1016/j.ejmech.2021.113925|pmid=34742013|s2cid=239486242}}&lt;/ref&gt;
* [[w:Lupin limited|Lupin]] and [[w:AbbVie Inc.|AbbVie]] are developing MALT1 protease inhibitors, in phase 1 clinical trials.&lt;ref&gt;{{Cite journal|date=2019-02-04|title=India's first-in-class MALT1 blocker deal|journal=Nature Biotechnology|volume=37|issue=2|page=112|doi=10.1038/s41587-019-0026-1|pmid=30718871|doi-access=free|s2cid=59603303}}&lt;/ref&gt;
* Excentia has an allosteric inhibitor &lt;ref&gt;{{Cite journal|last=Gooyit|first=M.|last2=Payne|first2=A.|last3=Carvalheda|first3=C.|last4=Evans|first4=D.|last5=Radoux|first5=C.|last6=Richards|first6=S.|last7=Besnard|first7=J.|last8=Varzandeh|first8=S.|last9=Gavard|first9=A.|date=2023-10|title=832P Characterisation of EXS73565: A potent and selective MALT1 inhibitor with low drug-drug interaction risk and potential in lymphoma|url=https://linkinghub.elsevier.com/retrieve/pii/S0923753423015326|journal=Annals of Oncology|language=en|volume=34|pages=S546–S547|doi=10.1016/j.annonc.2023.09.696}}&lt;/ref&gt;

==== Natural products ====

*Analogs of [[w:Quinone|β-Lapachone]] have been identified as MALT1 protease inhibitors.&lt;ref&gt;{{cite journal|authors=Lim SM, Jeong Y, Lee S, Im H, Tae HS, Kim BG, Park HD, Park J, Hong S|date=November 2015|title=Identification of β-Lapachone Analogs as Novel MALT1 Inhibitors To Treat an Aggressive Subtype of Diffuse Large B-Cell Lymphoma|journal=Journal of Medicinal Chemistry|volume=58|issue=21|pages=8491–502|doi=10.1021/acs.jmedchem.5b01415|pmid=26496175}}&lt;/ref&gt;
* [[w:secondary metabolites|secondary metabolites]] (oxepinochromenones) from the fungus [[w:Pleosporales|Dictyosporium]] show MALT1 protease inhibitory activity.&lt;ref&gt;{{cite journal|authors=Tran TD, Wilson BA, Henrich CJ, Staudt LM, Krumpe LR, Smith EA, King J, Wendt KL, Stchigel AM, Miller AN, Cichewicz RH, O'Keefe BR, Gustafson KR|date=January 2019|title=Secondary Metabolites from the Fungus Dictyosporium sp. and Their MALT1 Inhibitory Activities|journal=Journal of Natural Products|volume=82|issue=1|pages=154–162|doi=10.1021/acs.jnatprod.8b00871|pmc=7462088|pmid=30600998|s2cid=58540523}}&lt;/ref&gt;

==== Unknown ====

*[[w:Vlaams Instituut voor Biotechnologie|VIB]] is developing MALT1 protease inhibitors in collaboration with the [[w:Leuven|Leuven]]-based spin-off Centre for Drug Design and Discovery (CD3) &lt;ref name=&quot;germinating&quot;&gt;{{Cite journal|last=Cain|first=Chris|date=2014-02-06|title=Germinating MALT1|journal=SciBX: Science-Business EXchange|volume=7|issue=5|pages=133|doi=10.1038/scibx.2014.133|doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://sciencebusiness.net/network-news/vib-cd3-and-galapagos-nv-enter-license-deal-development-malt1-inhibitors|title=VIB, CD3 and Galapagos NV enter license deal for the development of MALT1 inhibitors|work=Science{{!}}Business|access-date=2019-05-31}}&lt;/ref&gt;
* Chordia therapeutics is entering a clinical trial with a MALT1 protease inhibitor in 2020&lt;ref&gt;{{Cite web|url=https://www.chordiatherapeutics.com/en/science/malt1.html|title=MALT1 Inhibitor CTX-177|date=2024-10-23|access-date=2024-10-23}}&lt;/ref&gt;
* Monopteros has a MALT1 inhibitor against solid tumor cancer.&lt;ref&gt;{{Cite journal|last=Naing|first=A.|last2=Park|first2=J.C.|last3=Klempner|first3=S.J.|last4=Khan|first4=S.|last5=Pearson|first5=P.|last6=Lufkin|first6=J.|last7=Keller|first7=P.|last8=Youssoufian|first8=H.|last9=DeCillis|first9=A.|date=2023-10|title=1033P First-in-human study of MALT1 inhibitor MPT-0118: Results from monotherapy dose escalation in advanced or metastatic refractory solid tumors|url=https://linkinghub.elsevier.com/retrieve/pii/S0923753423030090|journal=Annals of Oncology|language=en|volume=34|pages=S627–S628|doi=10.1016/j.annonc.2023.09.2172}}&lt;/ref&gt;
* Shrödinger has a compound in phase 1 clinical trials against mantle cell lymphoma.&lt;ref&gt;{{Cite journal|last=Yin|first=Wu|last2=Ye|first2=Min|last3=Marshall|first3=Netonia|last4=Tan|first4=Joanne BL|last5=Paull|first5=Evan|last6=Nie|first6=Zhe|last7=Trzoss|first7=MIchael|last8=Krilov|first8=Goran|last9=Feng|first9=Shulu|date=2023-11-02|title=Sgr-1505 Is a Potent MALT1 Protease Inhibitor with a Potential Best-in-Class Profile|url=https://ashpublications.org/blood/article/142/Supplement%201/2997/500137/Sgr-1505-Is-a-Potent-MALT1-Protease-Inhibitor-with|journal=Blood|language=en|volume=142|issue=Supplement 1|pages=2997–2997|doi=10.1182/blood-2023-190985|issn=0006-4971}}&lt;/ref&gt;
* Exelixis has a &quot;CARD11-BCL10-MALT1&quot; complex inhibitor in phase 1 clinical trial.&lt;ref&gt;{{Cite journal|last=O'Neil|first=Bert H.|last2=Solis|first2=Willy|last3=Vidal-Cardenas|first3=Sofia|last4=Uttamsingh|first4=Shailaja|last5=Kumar|first5=Abhijeet|date=2022-11-15|title=A First-in-Human Phase 1 Dose Escalation and Expansion Study of the Oral CARD11-BCL10-MALT1 Signaling Pathway Inhibitor XL114 in Patients with Non-Hodgkin's Lymphoma|url=https://ashpublications.org/blood/article/140/Supplement%201/12055/489490/A-First-in-Human-Phase-1-Dose-Escalation-and|journal=Blood|language=en|volume=140|issue=Supplement 1|pages=12055–12056|doi=10.1182/blood-2022-167824|issn=0006-4971}}&lt;/ref&gt;
* Rheos medicines has a clinical trial of their compound against autoimmune disease.&lt;ref&gt;{{Cite journal|last=Biswas|first=Subhabrata|last2=Chalishazar|first2=Aditi|last3=Helou|first3=Ynes|last4=DiSpirito|first4=Joanna|last5=DeChristopher|first5=Brian|last6=Chatterjee|first6=Devin|last7=Merselis|first7=Leidy|last8=Vincent|first8=Benjamin|last9=Monroe|first9=John G.|date=2022-05-09|title=Pharmacological Inhibition of MALT1 Ameliorates Autoimmune Pathogenesis and Can Be Uncoupled From Effects on Regulatory T-Cells|url=https://www.frontiersin.org/articles/10.3389/fimmu.2022.875320/full|journal=Frontiers in Immunology|volume=13|doi=10.3389/fimmu.2022.875320|issn=1664-3224|pmc=PMC9125252|pmid=35615349}}&lt;/ref&gt;

== References ==
{{reflist}}</text>
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{{HTitle|General Information}}
[https://en.wikipedia.org/wiki/Guatemala Guatemala],officially the Republic of Guatemala,[b] is a country in Central America. It is bordered to the north and west by Mexico, to the northeast by Belize, to the east by Honduras, and to the southeast by El Salvador. It is bounded by the Pacific Ocean to the south and the Gulf of Honduras to the northeast. Guatemala is a highly diverse country, populated by a variety of ethnic, cultural, racial, and linguistic groups. Guatemala’s sole official language is Spanish, spoken by 93 percent of the population as either the first or second language. 
Twenty-one [https://eld.bl.uk/catalog/100031307120%200x000001 Mayan] languages are spoken, especially in rural areas, as well as two non-Mayan Amerindian languages.

{{HTitle|Incidence and Prevalence of Hearing Loss }}
Guatemala does not have prevalence or incidence studies for hearing loss or any other disability; however, they have one of the highest birthrates in Latin America. According to the National Institute of Statistics, it is 2.4%, which in 2016 meant that approximately 1000 babies were born every day. If conservative international rates are applied, Guatemala should expect an average of three babies born every day with some degree of hearing loss. [https://sonrisasqueescuchan.org.gt/quienes-somos/?lang=en].Results were later published in a peer-reviewed journal.&lt;ref name=&quot;Spreckley2020&quot; /&gt;

While direct studies on the epidemiology of hearing loss in Guatemala are lacking, information on key demographic factors that can play a role can shed some light on the possible burden of hearing loss for the country. Guatemala is a multi-ethnic country with major populations including Maya, Ladino, Xinca, and Garifuna. A genetic study of 133 families with non-syndromic deafness found that a founder variant in the GJB2 gene (c.131G&gt;A) is a common cause of deafness in Guatemala, and that carriers share a haplotype closely matching the Mayan population.&lt;ref&gt;{{Cite journal|last=Carranza|first=C.|last2=Menendez|first2=I.|last3=Herrera|first3=M.|last4=Castellanos|first4=P.|last5=Amado|first5=C.|last6=Maldonado|first6=F.|last7=Rosales|first7=L.|last8=Escobar|first8=N.|last9=Guerra|first9=M.|date=2016-04|title=A Mayan founder mutation is a common cause of deafness in Guatemala|url=https://onlinelibrary.wiley.com/doi/10.1111/cge.12676|journal=Clinical Genetics|language=en|volume=89|issue=4|pages=461–465|doi=10.1111/cge.12676|issn=0009-9163|pmc=5484753|pmid=26346709}}&lt;/ref&gt;   Socioeconomic status has been associated with hearing loss. A 2020 study conducted in Guatemala reported that the provision of hearing aids was shown to improve economic circumstances, quality of life, and mental health among affected individuals.&lt;ref&gt;{{Cite journal|last=Spreckley|first=Mark|last2=Macleod|first2=David|last3=González Trampe|first3=Brenda|last4=Smith|first4=Andrew|last5=Kuper|first5=Hannah|date=2020-05-15|title=Impact of Hearing Aids on Poverty, Quality of Life and Mental Health in Guatemala: Results of a before and after Study|url=https://www.mdpi.com/1660-4601/17/10/3470|journal=International Journal of Environmental Research and Public Health|language=en|volume=17|issue=10|pages=3470|doi=10.3390/ijerph17103470|issn=1660-4601|pmc=7277678|pmid=32429252}}&lt;/ref&gt;  The availability and quality of audiological services in Guatemala are limited (see Challenges below). There is a lack of culturally sensitive client-family counseling and community-based counseling services, which are crucial for addressing the needs of those with hearing impairment. &lt;ref&gt;{{Cite book|title=Audiology in developing countries|date=2008|publisher=Nova Science Publishers|isbn=978-1-60456-945-2|editor-last=McPherson|editor-first=Bradley|location=New York|editor-last2=Brouillette|editor-first2=Ron}}&lt;/ref&gt;

{{HTitle|History of Audiology}}
Audiology services are present in Guatemala; however, there is only one university-trained audiologist in the country. Approximately 40 years ago, the first audiology shop started dispensing hearing aids; no diagnostic or repair services were available. Some years later, a second shop opened its doors. This one had better technology and more options for the clients.

As for pediatric services, hearing aids were being fitted without proper diagnosis; ABRs were being done by neurologists, and no behavioral observation techniques were used. Hearing aids were mainly body-worn aids and BTEs; very few people would try custom hearing aids, which were locally done. Digital technology was still years away.

In 1990, a new clinic opened its doors, led by a young professional with a Master’s degree in Education for the Deaf from [https://gallaudet.edu/ Gallaudet University]. This clinic offered top-of-the-line hearing aids, which were assembled in the United States. This seemed to be a trend for users who were looking for something new. This clinic, Centro de Audición, CEDAF, rapidly became the leader in hearing services and was responsible for introducing new technology through the years. Some examples are: the first completely in the canal (CIC) hearing aid line; the first digital aids; otoacoustic emissions (OAEs); Auditory Steady State Response (ASSR); cochlear implants; bone-anchored and middle ear implants; vestibular diagnosis (with equipment); real-ear measurements; and behavioral observation audiometry (with all its variants). During this time, the clinician pursued further studies and completed her Doctor of Audiology (Au.D.) at the [https://www.ufl.edu/ University of Florida]. 

After CEDAF came a few other clinics, most of which were opened by former employees of audiologic clinics. They had some training, which they received during their prior employment. 
To date, there are approximately 10 hearing clinics throughout the country. Except for two, all of them offer hearing aid fitting and pure tone audiometry. One offers: hearing aid fitting, auditory brainstem response (ABR), pure tone audiometry (PT), and recently vestibular testing. CEDAF offers: neonatal and school hearing screening; diagnostics for all ages (PT, ABR, and ASSR); all types of implants; vestibular diagnosis and rehabilitation; and hearing, speech, and language therapy.

{{HTitle|Information About Audiology}}
=== Education ===
No programs are available in Guatemala for audiology training. Speech and language therapy is available as an undergraduate program at a public university.

=== Audiology Practice: Public and Private ===
Public health systems, through Social Security, offer diagnostic services through contracts with private practitioners. Hearing aids are given to children who have completed their diagnostic tests before age five and whose parents are working either in the private or public sector. Adults may receive hearing aids if they are prescribed before retirement and it is not a congenital type of loss. Hearing aids are very basic, for both children and adults. All these services are free of charge. Cochlear implants are not available through public health services.

==== Private Sector ====
Most, if not all, audiological services are available through private clinics. There are differences in prices and quality among clinics.

==== Non-profit Organizations ====

* [https://sonrisasqueescuchan.org.gt/quienes-somos/?lang=en] (Smiles that Listen Foundation), which started in 2009. To date, it offers all services listed in the table below, except for cochlear implants.
* [https://prociegosysordos.org.gt/ Benemérito Comité Pro Ciegos y Sordos]: is a non-profit that serves people who are blind and/or deaf. They offer educational, medical, and rehabilitation services. Within the medical services they offer the following audiological services: ABR, pure tone audiometry, tympanometry and hearing aid fitting (very basic technology).

Another program that does not offer services but instead collects money to support cochlear implantation is [https://fundacionayudameaescuchar.org/ Ayúdame a Escuchar]. It started in 2014; however, a drawback is that they do not have audiology professionals locally, so they bring in an ear, nose, and throat (ENT) surgeon for each surgery, and an audiologist comes twice a year to map the implants. There are no spare parts or distributor of the brand in Guatemala.

=== Audiologic Services ===
{| class=&quot;wikitable&quot;
|+
!Service
!Clinics A 
(Cost in US Dollars)
!Clinics B 
(Cost in US Dollars)
!Approximate number of centers offering the service
!Service available in Public Sector
|-
|Neonatal Hearing Screening
|15-60
| -
|3
|Yes*
|-
|Pure Tone/ Speech Audiometry
|20-35
|10-18
|14
|Yes
|-
|Tympanometry
|20-35
|10-15
|13
|Yes
|-
|ABR/ASSR
|200-340
| -
|8
|Yes*
|-
|Hearing Aids
|800-3,500
|200-700
|14
|Yes*
|-
|Cochlear Implants
|24,000-28,000
| -
|1
|No
|-
|Vestibular Diagnosis
|200-275
| -
|2
|Yes*
|-
|Auditory Verbal Therapy
|20-30
| -
|1
|No
|-
|Speech/Language Therapy
|18-25
|10-15
|50
|No
|-
| colspan=&quot;5&quot; |• (*) Patients are sent to private practitioners who are subcontracted by the Public Health System due to lack of personnel and equipment.
• Clinics A target medium high and high income population

• Clinics B target medium and medium low-income population; they are less equipped and usually in peripheral areas.
|}

=== Services offered by Otolaryngologists, Otologist’s and Otoneurologists ===

==== Otolaryngologists ====

Medical and surgical care of patients with diseases and disorders that affect the ears, the respiratory and upper alimentary systems, and related structures of the head and neck: 
* Larynx and Nasal Endoscopy
* Few cochlear implants
* Ear wax removal
* Endoscopic Surgery
* Head and Neck Surgery
* Pediatric Otorhinolaryngology
* Audiometry
* Tympanometry
* Snoring Treatment
* Vestibular management
* Middle ear diagnosis and treatment

==== Otologists ====
There are no otologists per se; only four ENTs have skills in micro-ear surgery. Two of them have performed cochlear and bone anchored implants; however, one of them is now retired. Two more are in the process of developing these skills.

==== Otoneurologists ====
None at this time.

=== Audiological Services ===
{| class=&quot;wikitable&quot;
|+
!Service
!Approximate number of private centers offering the service
!Approximate number of centers in the Capital City
!Approximate number of centers in Rural Areas
|-
|Neonatal Hearing Screening
|3
|3
|0
|-
|Pure Tone/ Speech Audiometry
|14
|8
|6
|-
|Tympanometry
|13
|8
|5
|-
|ABR/ASSR
|8
|6
|2
|-
|Hearing Aids
|14
|8
|6
|-
|Cochlear Implants
|1
|1
|0
|-
|Vestibular Diagnosis
|2
|2
|0
|-
|Speech/ Language Therapy
|50
|40
|10
|-
|Auditory Verbal Therapy
|1
|1
|0
|-
|Cerumen Management
|Many
|Many
|Many
|-
|Intraoperative Monitoring
|1
|1
|0
|-
|Auditory Processing Disorders
|1
|1
|0
|-
| colspan=&quot;4&quot; |These services are offered by one Audiologist, audio technicians, and neurologists, ENT, psychologists and others.
|}

=== Professionals ===
{| class=&quot;wikitable&quot;
|+
!Professionals
!Approximate Number
!Ratio of the Population
|-
|Audiologists
|1
|16,000,000:1
|-
|Otolaryngologists
|46
|351,655:1
|-
|Micro-Ear Surgeons
|3
|5,400,000:1
|-
|Neurotologists
|0
| -
|-
|Speech-Language Pathologists
|400
|40,000:1
|-
|Teachers of the Deaf*
|60
|266,667:1
|-
|Hearing Aid Specialists
|40
|404,400:1
|-
| colspan=&quot;3&quot; |* Number of teachers with a bachelor’s degree in Education of Deaf Children. There are other teachers who do not have specialized education in the field. They are not accounted for. 
|}

=== Professional and Regulatory Bodies ===
There are no audiology professional organizations or regulatory agencies in Guatemala.

=== Scope of Practice and Licensing ===
There are no license requirements for audiology practitioners because there are no regulations for audiologists or training programs. Audiology practitioners basically do what their skills allow them to.

{{HTitle|Research in Audiology}}
In December 2016 the impact study among the low-income population in Guatemala xss([https://www.lshtm.ac.uk/sites/default/files/2019-06/do-hearing-aids-improve-lives.pdf Do hearing aids improve lives?]) was published by the London School of Hygiene &amp; Tropical Medicine, [https://www.wwhearing.org/guatemala World Wide Hearing] Foundation, and the International Centre for Evidence in Disability. 

{{HTitle|Audiology Charities}}
=== Fundación Sonrisas que Escuchan / Smiles that Listen Foundation ===
The foundation started in 2009 and is basically sponsored by a private practice, Centro de Audición, CEDAF. They share infrastructure, personnel, equipment, etc., therefore offering quality services at affordable prices. They offer: neonatal hearing screening, pure-tone, speech audiometry, tympanometry, ABR, ASSR, hearing aids, ear molds, hearing aid repair, etc. Services are not given free of charge; a social worker will set the contribution fee on an individual basis.

=== Benemérito Comité Pro Ciegos y Sordos ===
This is a non-profit that serves people who are blind and deaf. They offer educational, medical, and rehabilitation services. Within the medical services, they offer the following audiological services: ABR, pure-tone audiometry, tympanometry, and hearing aid fitting (very basic technology).

{{HTitle|Challenges, Opportunities and Notes}}
=== Challenges ===

* Audiology is unknown to many professionals
* Few professionals keep up to date in diagnosis, intervention and indication of HA fitting
* Little credibility and trust in audiology centers due to poor professional training
* In general, lack of information of where to seek help
* Audiological services are centralized in few cities, therefore transportation costs are involved
* Often, elderly are not brought for intervention, since their hearing loss seems to be “age appropriate”
* Hearing loss is usually not in government`s health care priorities
* Few trained people in the field of audiology
* No Audiology training programs available
* Lack of sufficient trained professionals, opens space to unethical “humanitarian missions”
* Calibration services are unavailable
* Few comprehensive audiological services
* Pediatric Audiological diagnostics are being done by professionals of other fields
* Insufficient specialized equipment
* Cost of diagnostic tests 
* High cost of the hearing aids
* User´s manual usually not in mother language
* Use of refurbished or used hearing aids without post-care services

{{HTitle|References}}
{{reflist}}


{{:Global Audiology/Authors-1|Dr. Patricia Castellanos de Muñoz|https://medicosenguatemala.com/medicos-cirujanos/otorrinolaringologos/dra-patricia-castellanos-de-munoz_17.html}}

Carranza C, Menendez I, Herrera M, Castellanos P, Amado C, Maldonado F, Rosales L, Escobar N, Guerra M, Alvarez D, Foster J 2nd, Guo S, Blanton SH, Bademci G, Tekin M. A Mayan founder mutation is a common cause of deafness in Guatemala. Clin Genet. 2016 Apr;89(4):461-465. doi: 10.1111/cge.12676. Epub 2015 Oct 6. PMID: 26346709; PMCID: PMC5484753.

''Edited in part by'' [http://linkedin.com/in/joyce-rodvie-sagun-4691bb182 Joyce Rodvie Sagun]

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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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      <comment>/* Visualizing the 120-cell */</comment>
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      <text bytes="88479" sha1="oajuoi1l8tosunpfptpapoghnostyxm" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - October 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>oajuoi1l8tosunpfptpapoghnostyxm</sha1>
    </revision>
    <revision>
      <id>2844988</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="88491" sha1="is1kw6r5y2utorzrby1bmgg38i5a4ro" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - October 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>is1kw6r5y2utorzrby1bmgg38i5a4ro</sha1>
    </revision>
    <revision>
      <id>2844993</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 16-cell 4-orthoplex */</comment>
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      <text bytes="88500" sha1="awsgeo3omo42pcxvaw6l7qozocdgngs" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - October 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="88501" sha1="fsk56k7pkocndybuq9l7ojk03hb7dxa" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <id>2844996</id>
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      <timestamp>2026-11-09T18:23:14Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Golden chords of the 120-cell */</comment>
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      <text bytes="88497" sha1="mue9l3tr8m10hawe0y70lro8jp7a35c" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
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|-
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>mue9l3tr8m10hawe0y70lro8jp7a35c</sha1>
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    <revision>
      <id>2844998</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 16-cell 4-orthoplex */</comment>
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      <text bytes="88498" sha1="s0bd9vle9xs4pnp347o4l6b2qt15zvu" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Every chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="88507" sha1="li5q1o0m2tjwc74e19vdyxhsxiwow87" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important rational numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Every chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>li5q1o0m2tjwc74e19vdyxhsxiwow87</sha1>
    </revision>
    <revision>
      <id>2845005</id>
      <parentid>2845004</parentid>
      <timestamp>2026-11-09T19:12:39Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="88498" sha1="s0bd9vle9xs4pnp347o4l6b2qt15zvu" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Every chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>s0bd9vle9xs4pnp347o4l6b2qt15zvu</sha1>
    </revision>
    <revision>
      <id>2845008</id>
      <parentid>2845005</parentid>
      <timestamp>2026-11-09T19:48:19Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="88866" sha1="kn4c13r2arwgq2uttesmnw41cdt5ito" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

Each of the 30 chords is the edge length of a distinct regular star polygon, a symmetry expression characterized by a distinct natural number. But the 30 chords do not form a known sequence of regular star polygons. The sequences of regular polygons generally have been shown to be mysteriously quasiperiodic and chaotic by the researchers cited in the introduction.

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Every chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>kn4c13r2arwgq2uttesmnw41cdt5ito</sha1>
    </revision>
    <revision>
      <id>2845009</id>
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      <contributor>
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      <text bytes="88867" sha1="4xlfobzefdsvc15reqawpil1ecqgd8v" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

Each of the 30 chords is the edge length of a distinct regular star polygon, a symmetry expression characterized by a distinct rational number. But the 30 chords do not form a known sequence of regular star polygons. The sequences of regular polygons generally have been shown to be mysteriously quasiperiodic and chaotic by the researchers cited in the introduction.

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Every chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>4xlfobzefdsvc15reqawpil1ecqgd8v</sha1>
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      <id>2845011</id>
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      <text bytes="88869" sha1="qdsfbtxrs0wpyvyy4gr6b83sf9snqsi" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

Each of the 30 chords is the edge length of a distinct regular star polygon, a symmetry expression characterized by a distinct rational number. But the 30 chords do not form a known sequence of regular star polygons. The sequences of regular polygons generally have been shown to be mysteriously quasiperiodic and chaotic by the researchers cited in the introduction.

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Every chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}, two completely orthogonal great squares which lie Clifford parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>qdsfbtxrs0wpyvyy4gr6b83sf9snqsi</sha1>
    </revision>
    <revision>
      <id>2845013</id>
      <parentid>2845011</parentid>
      <timestamp>2026-11-09T19:59:09Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 16-cell 4-orthoplex */</comment>
      <origin>2845013</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="88861" sha1="cks3tif9kc1u0l9etfdtd7khv2duui3" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - November 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;math&gt;\{5,3,3\}&lt;/math&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell) 4-orthoplex, not the 5-point (5-cell) 4-simplex, is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|-
|colspan=2|&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;
|&lt;math&gt;2.854102&lt;/math&gt;
|}

Each of the 30 chords is the edge length of a distinct regular star polygon, a symmetry expression characterized by a distinct rational number. But the 30 chords do not form a known sequence of regular star polygons. The sequences of regular polygons generally have been shown to be mysteriously quasiperiodic and chaotic by the researchers cited in the introduction.

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{3,3,4\}&lt;/math&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Every chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;math&gt;\{3,3,4\}&lt;/math&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}, two completely orthogonal great squares which lie Clifford parallel and perpendicular to each other. A ''simple rotation'' of the 16-cell in one of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double rotation'' of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles (an isoclinic rotation), moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, a hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;math&gt;\{4,3,3\}&lt;/math&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;math&gt;\{4,3,3\}&lt;/math&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with parallel effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding pairs of vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding pairs of vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagon central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagon central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,4,3\}&lt;/math&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;math&gt;\{3,4,3\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three completely disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}. The unit-radius planar {12}-gon has chords of length:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

The &lt;math&gt;r_1=\sqrt{1}&lt;/math&gt; chords form 8 Petrie dodecagons which zig-zag back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords. The 8 Petrie dodecagons can be divided four ways into 2 disjoint Petrie dodecagons {24/2}=2{12}.

The &lt;math&gt;r_2=\sqrt{1}&lt;/math&gt; chords form 16 great hexagons, which can be divided four ways into 4 Clifford parallel great hexagons {24/4}=4{6}.

The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chords form 18 great squares, which can be divided three ways into 6 Clifford parallel great squares {24/6}=6{4}, including one pair of completely orthogonal great squares from each of the three 16-cells.

The &lt;math&gt;r_4=\sqrt{3}&lt;/math&gt; chords form 32 great triangles, which can be divided four ways into 8 disjoint great triangles {24/8}=8{3} inscribed in 4 Clifford parallel great hexagons.

The &lt;math&gt;r_5=\sqrt{3}&lt;/math&gt; chords form 8 circular helix Clifford polygons, visible as a green {12/5} dodecagram in the orthogonal projection. An isoclinic rotation of the 24-cell in 4 invariant &lt;math&gt;r_2&lt;/math&gt; hexagon planes moves the vertices along 2 Clifford parallel circular isoclines {24/2}=2{12/5} over &lt;math&gt;r_5&lt;/math&gt; chords.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} shows three octagram isoclines of &lt;math&gt;\sqrt{2}&lt;/math&gt; chords in the 24-cell]]

We can rotate the 24-cell isoclinically in 6 Clifford parallel invariant great square planes containing 16-cell edges, in the great square rotation characteristic of the 16-cell, with parallel effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;math&gt;\sqrt{2}&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;math&gt;\sqrt{3}&lt;/math&gt; chords in the 24-cell]]We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing 24-cell edges, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon, which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A 24-cell great hexagon invariant plane revolution requires 720° like a 16-cell great square invariant plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in a great hexagon invariant plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as 4 great hexagon invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 12 &lt;math&gt;r_5&lt;/math&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. The rotational curve over each &lt;math&gt;r_5&lt;/math&gt; 120° chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Short chords
!Invariant planes
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|165°
|15°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|150°
|30°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|135°
|45°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;{24/12}=12{2}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|105°
|75°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we have found two distinct isoclinic rotations, the great square rotation characteristic of the 16-cell and the great hexagon rotation characteristic of the 24-cell. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these, and also four other distinct isoclinic rotations.  

Each row of the table describes a distinct isoclinic rotation of the 24-cell characterized by a pair of chords whose arc-lengths sum to 180°. Each chord lies in a central plane which is either a great square or a great hexagon. Each short chord plane is completely orthogonal to a corresponding long chord plane. These central planes are not to be confused with the invariant planes of the rotation, which intersect 0, 2, 4, or 6 vertices of the 24-cell as illustrated in the center column of each row.   

The short chord and long chord each have their characteristic {24/''n''}-gon, which correspond as projections of the 24-cell to completely orthogonal planes. Their projection viewpoints look straight down orthogonal cylinders in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; which are [[w:SO(4)#Visualization_of_4D_rotations|bent into tori]] in &lt;math&gt;\mathbb{R}^4&lt;/math&gt;.    

Each {24/''n''}-gon forms either a compound of ''n'' disjoint Clifford parallel regular polygons, or a single regular {24/n} star polygon. Polygons with {2}, {3}, {4} or {6} sides lie in a central plane, and all others lie skew in 4-space. The rotational angle between successive short chords in 4-space and the rotational angle between successive long chords in 4-space sum to 180°. Those angles distinguish distinct chords &lt;math&gt;t_i&lt;/math&gt; which are the same length.   

Each isoclinic rotation takes two chiral forms. There is a ''right rotation'' and a ''left rotation'' for each row of the table. A pair of right and left rotations are enantiomorphous reflections of each other, like a pair of clasped hands. The rotation takes Clifford parallel short chord polygons to each other, while the long chord polygons remain stationary in 4-space as vertices circle over them. The right and left rotations have non-congruent vertex position sequences: they take the short chord polygons through different sequences of short chord polygons.

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;math&gt;\{3,3,5\}&lt;/math&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the [[#The 24-cell|24-cell (see illustration above)]], which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{3,3,5\}&lt;/math&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt; the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has chords of length:

:&lt;math&gt;r_1=2 \times \sin(\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \times \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \times \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \times \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \times \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \times \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \times \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \times \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \times \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|96°
|84°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a distinct isoclinic rotation of the 600-cell, which occurs in right and left chiral forms. In both forms the long chord is the isocline chord of the stationary Clifford polygon over which vertices circle in 4-space.

Each distinct pair of complementary chord lengths is also identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]] where the lateral edge length is the chordal distance to the base polyhedron section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;math&gt;\sqrt{2}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell, with parallel effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This ''great square rotation characteristic of the 600-cell'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;math&gt;\sqrt{3}&lt;/math&gt;]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with parallel effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This ''great hexagon rotation characteristic of the 600-cell'' has 4 parallel orbits of period 30. The {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a 24-cell. The vertices in the invariant planes of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once.

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''great pentagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This rotation has 4 parallel orbits of period 30. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== Finally the 120-cell ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chords
! Section
! colspan=&quot;3&quot; |Long chords
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|180°
|0°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: palegreen;&quot; |
|168°
|12°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: gainsboro;&quot; |
|156°
|24°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|144°
|36°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|138.6~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: gainsboro;&quot; |
|138°
|42°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: palegreen;&quot; |
|132°
|48°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|130.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro;&quot; |
|128°
|52°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|124°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: gainsboro;&quot; |
|124°
|56°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|120°
|60°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|113.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro;&quot; |
|116°
|64°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|110.2~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: gainsboro;&quot; |
|112°
|68°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|108°
|72°
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: palegreen;&quot; |
|96°
|84°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|98.9~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
|95.5~°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: gainsboro;&quot; |
|°
|°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|90°
|90°
|}

The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;math&gt;\{5,3,3\}&lt;/math&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. 

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of 30 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. He showed that in spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance in &lt;math&gt;\mathbb{S}^3&lt;/math&gt; and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). In the 120-cell, each section also lies completely orthogonal to another congruent section.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

Only 8 of the 30 chords in the 120-cell occur in the 600-cell. The 120-cell's additional chords arise originally from the regular 5-cell 4-simplex, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is the entirety of what's new in the 120-cell.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of reflections in ''n'' dimensions. Application of the Fontaine and Hurley procedure to the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>cks3tif9kc1u0l9etfdtd7khv2duui3</sha1>
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  </page>
  <page>
    <title>Patriarch Ages Curious Numerical Facts Response</title>
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    <id>328204</id>
    <revision>
      <id>2845056</id>
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      <timestamp>2026-11-10T03:22:24Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
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      <text bytes="128064" sha1="5iyjiekghd3cj1rhyl4kz4nnhqxg1pq" xml:space="preserve">{| class=table style=&quot;width:100%;&quot;
|-
 | {{Original research}}
|-
 | {{RoundBoxTop}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional context and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. This article was researched and written by a human author, with editing assistance provided by [https://gemini.google.com Google &quot;Gemini, your personal AI assistant&quot;]{{RoundBoxBottom}}
|}

= The Sumerian Origins of Genesis =
The early history in Genesis shares deep structural, thematic, and chronological ties with ancient Mesopotamian writing traditions. There is a close parallel between the biblical figures listed in Genesis 5, who lived before the flood, and those listed in the '''[[w:Sumerian King List|Sumerian King List (SKL)]]'''. This ancient text details a succession of long-lived rulers who reigned right up until a cataclysmic flood destroyed their civilization. The table below shows the connections between Genesis 5 and the Sumerian King List, with directly matching details highlighted in blue.

{| class=&quot;nowraplinks&quot; cellspacing=&quot;0&quot; cellpadding=&quot;3&quot; rules=&quot;all&quot; style=&quot;background:#fbfbfb; border: 1px solid #ccc; font-size:90%; empty-cells:show; border-collapse:collapse; text-align:left;&quot;
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! colspan=&quot;2&quot; | Sumerian King List
! Hebrew Bible
|- style=&quot;text-align:center; font-style:italic;&quot;
! colspan=&quot;2&quot; style=&quot;background:#fffcf2;&quot; | Archaeological Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2400&lt;/span&gt; Years Total
! style=&quot;background:#fffcf2;&quot; | Demetrius Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2264&lt;/span&gt; Years Total
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! City
! Ruler (Sage)
! Patriarch
|-
| [[w:Eridu|Eridu]]
| [[w:Alulim|Alulim]] &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;([[w:Adapa|Adapa]])&lt;/span&gt; &lt;br /&gt; [[w:Alalngar|Alalngar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Adam|Adam]]&lt;/span&gt; &lt;br /&gt; [[w:Seth|Seth]]
|-
| [[w:Bad-tibira|Bad-tibira]]
| [[w:En-men-lu-ana|En-men-lu-ana]] &lt;br /&gt; [[w:En-men-gal-ana|En-men-gal-ana]] &lt;br /&gt; [[w:Dumuzid|Dumuzid]] 
| [[w:Enos|Enos]] &lt;br /&gt; [[w:Kenan|Kenan]] &lt;br /&gt; [[w:Mahalalel|Mahalalel]]
|-
| [[w:Larak (Sumer)|Larak]] 
| [[w:En-sipad-zid-ana|En-sipad-zid-ana]] 
| [[w:Jared (biblical figure)|Jared]]
|-
| [[w:Sippar|Sippar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:En-men-dur-ana|En-men-dur-ana]]&lt;/span&gt; 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Enoch|Enoch]]&lt;/span&gt;
|-

| [[w:Shuruppak|Shuruppak]]
| [[w:Ubara-Tutu|Ubara-Tutu]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Utnapishtim|Utnapishtim]]&lt;/span&gt; 
| [[w:Methuselah|Methuselah]] &lt;br /&gt; [[w:Lamech (father of Noah)|Lamech]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Noah|Noah]]&lt;/span&gt;
|}

== Narrative Similarities ==
The biblical accounts seem to build on, or develop alongside, the legendary roles given to Sumerian rulers and their wise advisors, known as apkallu (sages). Key thematic connections include:

*[https://www.worldhistory.org/article/216/the-myth-of-adapa/ Adam as Adapa] : In the Adapa narrative, Ea (god of wisdom) warns Adapa to refuse any food or drink offered in heaven but to accept clothing and oil. Anu (the sky god) offers Adapa food and drink, followed by clothing and oil. When Adapa refuses the food, Anu chides him for missing an opportunity for immortality. Similarly, in the Genesis account, Yahweh Elohim permits Adam and Eve to eat from any tree in the divine garden except the tree of knowledge. A competing figure, the serpent, advises them to eat from it anyway. After they do, Yahweh returns, chastises them for disobedience, and provides garments of skin. Key parallels between the accounts include: A potential similarity and shared root in the names Adapa and Adam. Both narratives feature a test involving the consumption of supposedly deadly food. Both figures are offered clothing and summoned before a deity to answer for their decisions.

*[https://www.marquette.edu/maqom/Chapter%201.pdf Enoch as Enmeduranki]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew Book of Enoch describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text Pirišti Šamê u Erṣeti (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.

*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

== Chronological Similarities ==
[[File:Antediluvian Sumerian King List Cities and Biblical Rivers.png|thumb|400px|Locations of four of the five pre-Flood cities named in the Sumerian King List have been identified through excavated cuneiform inscriptions. These cities existed in close proximity to the Tigris and Euphrates rivers. Eridu was a coastal city when first founded around 5300 BC. Over the millennia, however, massive deposits of alluvial silt from four river systems filled the ancient bays, gradually pushing the shoreline southeastward away from Eridu. There are four rivers named in the biblical Garden of Eden story (Genesis 2:10–14): the Tigris, Euphrates, Gihon, and Pishon. The Gihon and Pishon are frequently associated with the Karun River in western Iran and the Wadi al-Batin (a now-dry riverbed) in the Arabian Peninsula. The blue oval on the map above represents the 5300 BC coastline, with arrows pointing to the mouths of the four Eden rivers.]]

In Mesopotamian mythology, Eridu was the oldest city, the home of the wisdom god Enki, and the first site where divine kingship descended. While many Sumerian texts omit the pre-Flood era entirely, in narratives that do include a pre-Flood prologue, Eridu is consistently identified as the first city and Šuruppak is frequently identified as the fifth and final city prior to the Flood.

One English translation of the Sumerian King List states it as follows: [https://www.livius.org/sources/content/anet/266-the-sumerian-king-list/ &quot;After the kingship descended from heaven, the kingship was in Eridu ... Bad-tibira fell and the kingship was taken to Larak ... Sippar fell and the kingship was taken to Šuruppak ... Five cities; eight kings ruled for 385,200 years. Then the Flood swept over.&quot;]

A related text, the [[w:Eridu Genesis|Eridu Genesis]], preserves a parallel tradition with Eridu as the first city and Šuruppak as the fifth: [https://www.livius.org/sources/content/oriental-varia/eridu-genesis/ &quot;The firstling of the cities, Eridu, she gave to the leader Nudimmud, the second, Bad-Tibira, she gave to the Prince and the Sacred One, the third, Larak, she gave to Pahilsag, the fourth, Sippar, she gave to the gallant Utu, the fifth, Šuruppak, she gave to Ansud.&quot;]

Locations of four of the five pre-Flood Sumerian cities have been identified archeologically. Eridu sits on a hill inside a basin roughly {{convert|15|mi}} long and {{convert|20|ft}} deep, separated from the Euphrates by a sandstone ridge called the ''Hazem''. In the 3rd Millennium BC, before the river shifted course, a canal connected Eridu to the Euphrates. Low tells with 2nd Millennium BC pottery and later burials mark the canal's path.

====Prehistoric Temple(s)====
[[File:Ea (Babilonian) - EnKi (Sumerian).jpg|thumb|Detail of the Adda Seal, an ancient Akkadian cylinder seal (circa 2300 BCE) depicting Enki with water streams coming from his shoulders.|270x270px]]
The urban nucleus of Eridu was Enki's temple, called the ''House of the Aquifer'' in the oldest Sumerian texts. In later texts, this temple was called the ''House of the Waters''. [[w:Enki|Enki]] was the Mesopotamian god of subterranean waters, specifically reigning over the primeval underground ocean known as the [[w:Abzu|Abzu]]. In ancient art, he was frequently represented with water streams flowing directly from his body, which were often filled with fish swimming through the currents. The biblical [[w:Garden of Eden|Garden of Eden]] is presented as a well-watered garden with rivers running out of it. The Sumerian association of Eridu with subterranean waters seems reflected in the biblical garden story.

[[File:Eridu Level Sequence A.jpg|frame|Prehistoric sequence of buildings excavated at Eridu.]]
A prehistoric sequence of buildings excavated at Eridu during the 1946 to 1949 campaigns seems to substantiate Eridu's status as the first city with the earliest temple. Consisting of eighteen occupational levels, this Eridu sequence of &quot;Palace-like&quot; or &quot;Temple-Like&quot; buildings appears to be the first occurrence of such structures in the region&lt;ref&gt;{{cite book| last= Leick| first= Gwendolyn| title= Mesopotamia, The invention of the city| place= London| publisher= Penguin Books| year= 2002}}&lt;/ref&gt;.

In [[w:Archaeology|archaeology]], a '''[[w:Tell (archaeology)|tell]]''' is a mound consisting of the accumulated and stratified debris of a succession of consecutive settlements at the same site, the refuse of generations of people who built and inhabited them, alongside natural sediment. The most recent layers of a tell are nearest to the surface. As archaeologists dig into the tell, they encounter successively older layers of civilization, a recording of history called [[w:Stratigraphy|stratigraphy]]. 

The following table compares the stratigraphy found during the Eridu 1946–1949 campaigns with the stratigraphy found during the 1931 excavation of Shuruppak (Tell Fara). The table is organized according to the order in which layers are uncovered, with the youngest layers in the first rows of the table and successively older layers in subsequent rows. As shown below, the historic timeline separating the first construction at Eridu (c. 5300 BC) from the cultural break of the Shuruppak flood (c. 2900 BC) spans a real-world duration of approximately 2,400 years.

{| class=&quot;wikitable&quot;
! Historical Period &amp; Dating
! Eridu Temple Stratigraphy
! Shuruppak Stratigraphy
|-
| [[w:Sumu-abum|Amorite Era]]
|-
| ''' Dynastic Periods '''
''(c. 2004-2900 BC)''
| '''Ur III Ziggurat Layer''' &lt;br/&gt; &quot;In the ruins of the ziggurat we found numerous bricks of&quot; [[w:Amar-Sin|Amar-Sin (c. 2046–2037 BC)]], &quot;three bricks of&quot; [[w:Ur-Nammu|Ur-Nammu (2112–2094 BC)]] &quot;one brick of&quot; [[w:Nur-Adad|Nur-Adad (c. 1866-1850 BC)]] &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;It seems that Ur-Nammu began the building of this ziggurat, but the bulk of the task was left to his grandson Amar-Sin to accomplish.&quot;]
| '''Fara II &amp; III Layer'''
([[w:Shu-Sin|Shu-Sin]] inscription (c. 2037 – 2028 BC), Sumerian seal cylinders, pottery, skeletons, figurines, Globular Stone Jar)
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 2900 BC'''
| style=&quot;text-align: center;&quot; | –
| style=&quot;background-color:#ffdddd;&quot; | '''The Shuruppak Flood Layer'''
[https://www.penn.museum/sites/journal/9356/ &quot;We found an alluvial layer at a depth between four and five metres below the mound surface.&quot;] [https://www.penn.museum/sites/journal/9356/ &quot;Only a few objects of this period, namely one stamp seal and some sherds “floated” in the refuse above. In this respect an absolute culture break took place.&quot;]
|-
| '''[[w:Jemdet Nasr period|Jemdet Nasr]]'''
''(c. 2900-3100 BC)''
| style=&quot;text-align: center;&quot; | –
| '''Fara I Layer''' 
(Scaraboid Stamp Seals, Polychrome Jar, Animal Patterns on Potsherds, Copper Blade)
|-
| '''[[w:Uruk period|Uruk]]'''
''(c. 3100-3800 BC)''
| '''Temple Layers I–V''' &lt;br/&gt; Temple I &amp; II: [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;All that is left to us are its foundations and retaining-walls, while even of these a large portion are buried beneath the massive masonry of the Third Dynasty Ziggurat.] &lt;br/&gt; Temple III: Small bricks of reddish clay. &lt;br/&gt; Temple IV: Medium sized bricks of greenish clay. &lt;br/&gt; Temple V: Large bricks of light-colored clay with dark joints. &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;Since Temple VI was now known to belong to a very late, if not the final stage of the Al-Ubaid period, Temples III, IV and V must be taken to represent a stage intermediate between this and the stone architecture of the Protoliterate age&quot;]
| style=&quot;text-align: center;&quot; | –
|-
| '''Ubaid 1-4'''
''(c. 3800-5300 BC)''
| '''Temples VI–XVIII''' &lt;br/&gt; Temple X: &quot;At Level X the temple appeared to have been rebuilt with only minor changes in the plan.&quot; &lt;br/&gt; Temple XI: &quot;provides the first instance of a temple building being deliberately raised upon a platform.&quot; &lt;br/&gt; Temple XII &amp; XIII: &quot;there were two successive occupation-levels at which no traces of actual buildings were to be found&quot;  &lt;br/&gt; Temple XVIII &quot;The first traces of occupation at Eridu occurred at a level corresponding to 11.70 meters beneath datum&quot; ... &quot;and, directly above, at Level XVII, were the remains of the earliest recognizable building&quot;
| style=&quot;text-align: center;&quot; | –
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 5300 BC'''
| style=&quot;background-color:#ffdddd;&quot; | '''First Construction'''
| style=&quot;text-align: center;&quot; | –
|}

=== Pre-flood Bible Chronology ===
'''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal. Writing from [https://commons.wikimedia.org/wiki/File:Ptolemaic_Alexandria_-_en.svg Alexandria in the late 3rd century BC (c. 221 BC)], Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood of Noah.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;11&quot; style=&quot;background-color:#e8e8e8;&quot; | —
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 230
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 205
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 190
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 170
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1309
|}

The Septuagint chronology, as witnessed by Demetrius at 2,264 years, mirrors this archaeological reality within a remarkably tight margin of error (less than 6%).
This broad-stroke alignment strongly indicates that while the fine internal details were arranged to fit sacred numerical cycles, the overarching framework of the Septuagint preserved an authentic, historically grounded memory of the Fertile Crescent's true antiquity—a memory that was severely contracted in alternative textual traditions.

The structural overlap between the Mesopotamian and biblical traditions becomes most compelling when evaluating the total duration of the antediluvian era. While the fine details and specific individual lifespans in both the Septuagint and the Sumerian King List are heavily calculated using stylized, sacred, or mathematically perfect numbers, the macro-span of the Septuagint timeline displays an uncanny alignment with real-world ancient history.

This macro-alignment is particularly remarkable when contrasted against the Sumerian King List itself. In the Mesopotamian scribal tradition, the antediluvian era spans astronomical lengths of time—typically totaling over 240,000 years across the reigns of the first eight pre-flood kings. These numbers are clearly symbolic or mathematical inflations that completely diverge from physical archaeological realities.

Rather than duplicating these vast Near Eastern timelines or severely compressing the era, the Septuagint tradition preserved a highly realistic historical interval. 

It is highly significant that the Septuagint chronology, as witnessed by Demetrius, aligns so closely with modern archaeological horizons instead of the exaggerated spans of the SKL. Because an unguided mathematical arrangement of sacred numbers would be statistically unlikely to land so close to actual physical history, this uncanny broad-stroke accuracy hints at two intriguing possibilities: it was either a case of monumental cosmic luck, or the Alexandrian redactors possessed ancient, pristine Near Eastern source records that have since been entirely lost to history.

This total of 2,264 years stands in stark contrast to the significantly compressed timelines preserved in the Masoretic Text (1,656 years) and the Samaritan Pentateuch (1,307 years). When evaluating these numbers against physical history, the Septuagint's macro-chronology reveals a striking real-world correlation.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].


==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167 / 187
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

{{RoundBoxTop}}
====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.


Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.
{{RoundBoxBottom}}
{{RoundBoxTop}}
'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
{{RoundBoxBottom}}

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors, as shown in the following tables.

(The '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the Black Desert of Jordan. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

{{RoundBoxTop}}
==== The Birth of Shem (A Digression) ====
Were Noah's sons born when Noah was 500 or 502?

While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.
{{RoundBoxBottom}}

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 - 1}{2} + 3^2 &amp;= 1200 + 9 \\
 &amp;= 1209
\end{aligned}
&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 + 1}{2} + (7^2 - 3^2) &amp;= 1201 + 40 \\
 &amp;= 1241
\end{aligned}
&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== The Masoretic text Variation ==


[[File:Schematic_Diagram_Masoretic_Text_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Masoretic Text'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

=== Mathematical Structure of the Masoretic Chronology ===
The above diagram maps the Masoretic chronology using the identical grid framework applied to the Samaritan Pentateuch. Red circles indicate specific chronological points where the Masoretic timeline diverges from the Samaritan tradition. Key textual and numerical variations include:

==== The Universal Flood Adjustment (+349 yrs) ====
The Masoretic chronology postpones the birth of Noah and the onset of the Deluge by '''349 years''' relative to the Samaritan tradition. As outlined in the above &quot;Masoretic Adjustments&quot; section, this structural shift ensures that Noah's antediluvian ancestors do not survive past the date of the universal flood. This timeline was achieved by adding 100 years to Jared's paternal age, 120 years to Methuselah's, and 129 years to Lamech's. Consequently, these calculations place Methuselah's death precisely in the year of the Flood and Lamech's death five years prior.

==== The Smoothing of Noah's Life (-2 yrs) ====
The Masoretic chronology reduces Noah's age at the birth of Shem from 502 years (as found in the Samaritan Pentateuch) to 500 years. As discussed in the &quot;Lectio Difficilior Potior&quot; and &quot;The Birth of Shem (A Digression)&quot; sections, this change represents &quot;scribal smoothing.&quot; This editorial process rounded major milestones in Noah's life to the nearest century: his sons are born in his 500th year, the Deluge occurs in his 600th year, and his father Lamech dies in his 700th year.

==== Post-Flood Patriarchal Lifespans (-650 yrs) ====
Seven post-flood patriarchs (Arpachshad through Nahor) exhibit reduced ages at the birth of their firstborn sons within the Masoretic chronology, resulting in a cumulative deficit of 650 years compared to the Samaritan Pentateuch. Specifically, the paternal ages of Arpachshad, Shelah, Eber, Peleg, Reu, and Serug are each lower by exactly 100 years, while Nahor's age is reduced by 50 years. Given the uniform, rounded values of these variations, a direct dependency exists; either the Masoretic chronologist systematically deflated these ages, or the Samaritan chronologist systematically inflated them.

The Masoretic use of young fathering ages for these specific patriarchs may preserve an older tradition. The narrative theme of Abraham fathering Isaac at the age of 100 serves as a central literary and mathematical motif that likely predates the Babylonian exile. Consequently, the lower paternal ages in the Masoretic lineage may intentionally highlight this milestone, ensuring that a fatherhood age of 100 remains an extraordinary, miraculous exception within the broader genealogical timeline.

Under this proposed framework, Arpachshad (or Cainan) originally functioned not as a standard descendant of Shem, but as the progenitor of a distinct nine-generation patriarchal line culminating in the miraculous birth of Isaac. During the Babylonian exile, the Priestly source (&quot;P&quot;) may have adapted this older Abrahamic lineage, appending earlier primeval generations to mirror Mesopotamian creation and flood epics. Consequently, Mesopotamian archetypes—such as Adapa, Enmeduranki, and Utnapishtim—were synthesized with preexisting Hebrew traditions to form the composite figures of Adam, Enoch, and Noah. These composite primeval figures were subsequently integrated into a single, seamless genealogical chain spanning from creation to the foundations of Israel.

==== The Final Masoretic Adjustment (+215 yrs) ====
As will be explained later in this resource, it is proposed that a final 215 years was added to the Masoretic chronology to arrive at the 2666-year total. The Samaritan Pentateuch, the Septuagint, and the Book of Jubilees all claim that the Patriarchs were in Canaan for 215 years, followed by 215 years of Israelite slavery in Egypt. The Masoretic text increases the period of Egyptian slavery by 215 years, for a total of 430 years of slavery.

This increase to 430 years is difficult to reconcile with the Moses narrative, which claims that Moses was the great-grandson of Levi on the paternal line and the grandson on the maternal line. Levi entered Egypt as an adult and a free man. Moses led the children of Israel out of Egypt at eighty years old. For Moses to be a maternal grandson, his mother would have been hundreds of years old when she gave birth to him and his older siblings, Aaron and Miriam.

==== The 2800-Year to 2666-Year Transition ====
Within the Masoretic chronology, the Exodus is situated inside the third 800-year epoch, occurring precisely in the year 2666 after creation (Anno Mundi). This placement represents a deliberate scribal design choice rather than an organic chronological coincidence.

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;

* '''Masoretic Text (Adam to Exodus):'''
:&lt;math display=&quot;block&quot;&gt;
(2800 - 46) + (349) - (2) - (650) + (215) = 2666&lt;/math&gt;
&lt;/div&gt;

As detailed in &quot;The 350-Year Symmetrical Extension,&quot; the timeline found in the Book of Jubilees likely served as the foundational model, positioning Shem at the exact midpoint of its 2,450-year chronology. The Samaritan Pentateuch appears to be a direct derivative of this Jubilees framework. Scribes systematically appended exactly 350 years to the post-Flood era, shifting the completion of the Conquest to a precise macro-structural target of 2,800 years after creation.

Envisioning the process that transitioned from the 2,800-year Samaritan chronology to the Masoretic framework presents a distinct mathematical challenge. Tracking the adjustments step-by-step from the Samaritan baseline reveals a precise transition:

* '''Samaritan Baseline (Conquest Completion):''' 2,800 years
* '''Minus Wandering and Conquest Era:''' −46 years (2,754)
* '''Plus Masoretic Flood Adjustment:''' +349 years (3,103)
* '''Minus Noah's Life Smoothing:''' −2 years (3,101)
* '''Minus Post-Flood Patriarchal Reductions:''' −650 years (2,451)
* '''The Final Masoretic Adjustment:''' +215 years (2,666)

This calculation yields exactly 2,666 years. Because individual components were dictated by rigid external theological and narrative constraints, it is surprising that the final adjustment yields exactly 2,666 years, but it also remains difficult to isolate where scribes might have executed fine-tuning to bring about this exact value and bridge the 2,800-year and 2,666-year macro-structures.

=== Mathematical Structure of the Septuagint Chronology ===
As was explained above in &quot;Septuagint Adjustments&quot;, the LXX’s editor methodically added 100 years to each pre-flood patriarch. The purpose of this adjustment was to ensure that Noah would be the exclusive survivor of the flood. As shown in the tables below, post-flood &quot;Long Chronology&quot; variations are seen to be either identical to or derived from &quot;Short Chronology&quot; values.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. The [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology shown above, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.

[[Category:Religion]]


=== Mythology ===
In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.</text>
      <sha1>5iyjiekghd3cj1rhyl4kz4nnhqxg1pq</sha1>
    </revision>
    <revision>
      <id>2845058</id>
      <parentid>2845056</parentid>
      <timestamp>2026-11-10T03:26:26Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Prehistoric Temple(s) */</comment>
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      <text bytes="128192" sha1="1q95mq2e7m6ew8hlvmeqr6lb7ocmnah" xml:space="preserve">{| class=table style=&quot;width:100%;&quot;
|-
 | {{Original research}}
|-
 | {{RoundBoxTop}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional context and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. This article was researched and written by a human author, with editing assistance provided by [https://gemini.google.com Google &quot;Gemini, your personal AI assistant&quot;]{{RoundBoxBottom}}
|}

= The Sumerian Origins of Genesis =
The early history in Genesis shares deep structural, thematic, and chronological ties with ancient Mesopotamian writing traditions. There is a close parallel between the biblical figures listed in Genesis 5, who lived before the flood, and those listed in the '''[[w:Sumerian King List|Sumerian King List (SKL)]]'''. This ancient text details a succession of long-lived rulers who reigned right up until a cataclysmic flood destroyed their civilization. The table below shows the connections between Genesis 5 and the Sumerian King List, with directly matching details highlighted in blue.

{| class=&quot;nowraplinks&quot; cellspacing=&quot;0&quot; cellpadding=&quot;3&quot; rules=&quot;all&quot; style=&quot;background:#fbfbfb; border: 1px solid #ccc; font-size:90%; empty-cells:show; border-collapse:collapse; text-align:left;&quot;
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! colspan=&quot;2&quot; | Sumerian King List
! Hebrew Bible
|- style=&quot;text-align:center; font-style:italic;&quot;
! colspan=&quot;2&quot; style=&quot;background:#fffcf2;&quot; | Archaeological Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2400&lt;/span&gt; Years Total
! style=&quot;background:#fffcf2;&quot; | Demetrius Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2264&lt;/span&gt; Years Total
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! City
! Ruler (Sage)
! Patriarch
|-
| [[w:Eridu|Eridu]]
| [[w:Alulim|Alulim]] &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;([[w:Adapa|Adapa]])&lt;/span&gt; &lt;br /&gt; [[w:Alalngar|Alalngar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Adam|Adam]]&lt;/span&gt; &lt;br /&gt; [[w:Seth|Seth]]
|-
| [[w:Bad-tibira|Bad-tibira]]
| [[w:En-men-lu-ana|En-men-lu-ana]] &lt;br /&gt; [[w:En-men-gal-ana|En-men-gal-ana]] &lt;br /&gt; [[w:Dumuzid|Dumuzid]] 
| [[w:Enos|Enos]] &lt;br /&gt; [[w:Kenan|Kenan]] &lt;br /&gt; [[w:Mahalalel|Mahalalel]]
|-
| [[w:Larak (Sumer)|Larak]] 
| [[w:En-sipad-zid-ana|En-sipad-zid-ana]] 
| [[w:Jared (biblical figure)|Jared]]
|-
| [[w:Sippar|Sippar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:En-men-dur-ana|En-men-dur-ana]]&lt;/span&gt; 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Enoch|Enoch]]&lt;/span&gt;
|-

| [[w:Shuruppak|Shuruppak]]
| [[w:Ubara-Tutu|Ubara-Tutu]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Utnapishtim|Utnapishtim]]&lt;/span&gt; 
| [[w:Methuselah|Methuselah]] &lt;br /&gt; [[w:Lamech (father of Noah)|Lamech]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Noah|Noah]]&lt;/span&gt;
|}

== Narrative Similarities ==
The biblical accounts seem to build on, or develop alongside, the legendary roles given to Sumerian rulers and their wise advisors, known as apkallu (sages). Key thematic connections include:

*[https://www.worldhistory.org/article/216/the-myth-of-adapa/ Adam as Adapa] : In the Adapa narrative, Ea (god of wisdom) warns Adapa to refuse any food or drink offered in heaven but to accept clothing and oil. Anu (the sky god) offers Adapa food and drink, followed by clothing and oil. When Adapa refuses the food, Anu chides him for missing an opportunity for immortality. Similarly, in the Genesis account, Yahweh Elohim permits Adam and Eve to eat from any tree in the divine garden except the tree of knowledge. A competing figure, the serpent, advises them to eat from it anyway. After they do, Yahweh returns, chastises them for disobedience, and provides garments of skin. Key parallels between the accounts include: A potential similarity and shared root in the names Adapa and Adam. Both narratives feature a test involving the consumption of supposedly deadly food. Both figures are offered clothing and summoned before a deity to answer for their decisions.

*[https://www.marquette.edu/maqom/Chapter%201.pdf Enoch as Enmeduranki]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew Book of Enoch describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text Pirišti Šamê u Erṣeti (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.

*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

== Chronological Similarities ==
[[File:Antediluvian Sumerian King List Cities and Biblical Rivers.png|thumb|400px|Locations of four of the five pre-Flood cities named in the Sumerian King List have been identified through excavated cuneiform inscriptions. These cities existed in close proximity to the Tigris and Euphrates rivers. Eridu was a coastal city when first founded around 5300 BC. Over the millennia, however, massive deposits of alluvial silt from four river systems filled the ancient bays, gradually pushing the shoreline southeastward away from Eridu. There are four rivers named in the biblical Garden of Eden story (Genesis 2:10–14): the Tigris, Euphrates, Gihon, and Pishon. The Gihon and Pishon are frequently associated with the Karun River in western Iran and the Wadi al-Batin (a now-dry riverbed) in the Arabian Peninsula. The blue oval on the map above represents the 5300 BC coastline, with arrows pointing to the mouths of the four Eden rivers.]]

In Mesopotamian mythology, Eridu was the oldest city, the home of the wisdom god Enki, and the first site where divine kingship descended. While many Sumerian texts omit the pre-Flood era entirely, in narratives that do include a pre-Flood prologue, Eridu is consistently identified as the first city and Šuruppak is frequently identified as the fifth and final city prior to the Flood.

One English translation of the Sumerian King List states it as follows: [https://www.livius.org/sources/content/anet/266-the-sumerian-king-list/ &quot;After the kingship descended from heaven, the kingship was in Eridu ... Bad-tibira fell and the kingship was taken to Larak ... Sippar fell and the kingship was taken to Šuruppak ... Five cities; eight kings ruled for 385,200 years. Then the Flood swept over.&quot;]

A related text, the [[w:Eridu Genesis|Eridu Genesis]], preserves a parallel tradition with Eridu as the first city and Šuruppak as the fifth: [https://www.livius.org/sources/content/oriental-varia/eridu-genesis/ &quot;The firstling of the cities, Eridu, she gave to the leader Nudimmud, the second, Bad-Tibira, she gave to the Prince and the Sacred One, the third, Larak, she gave to Pahilsag, the fourth, Sippar, she gave to the gallant Utu, the fifth, Šuruppak, she gave to Ansud.&quot;]

Locations of four of the five pre-Flood Sumerian cities have been identified archeologically. Eridu sits on a hill inside a basin roughly {{convert|15|mi}} long and {{convert|20|ft}} deep, separated from the Euphrates by a sandstone ridge called the ''Hazem''. In the 3rd Millennium BC, before the river shifted course, a canal connected Eridu to the Euphrates. Low tells with 2nd Millennium BC pottery and later burials mark the canal's path.

====Prehistoric Temple(s)====
[[File:Ea (Babilonian) - EnKi (Sumerian).jpg|thumb|Detail of the Adda Seal, an ancient Akkadian cylinder seal (circa 2300 BCE) depicting Enki with water streams coming from his shoulders.|270x270px]]
The urban nucleus of Eridu was Enki's temple, called the ''House of the Aquifer'' in the oldest Sumerian texts. In later texts, this temple was called the ''House of the Waters''. [[w:Enki|Enki]] was the Mesopotamian god of subterranean waters, specifically reigning over the primeval underground ocean known as the [[w:Abzu|Abzu]]. In ancient art, he was frequently represented with water streams flowing directly from his body, which were often filled with fish swimming through the currents. The biblical [[w:Garden of Eden|Garden of Eden]] is presented as a well-watered garden with rivers running out of it. The Sumerian association of Eridu with subterranean waters seems reflected in the biblical garden story.

[[File:Eridu Level Sequence A.jpg|frame|Prehistoric sequence of buildings excavated at Eridu.]]
A prehistoric sequence of buildings excavated at Eridu during the 1946 to 1949 campaigns seems to substantiate Eridu's status as the first city with the earliest temple. Consisting of eighteen occupational levels, this Eridu sequence of &quot;Palace-like&quot; or &quot;Temple-Like&quot; buildings appears to be the first occurrence of such structures in the region&lt;ref&gt;{{cite book| last= Leick| first= Gwendolyn| title= Mesopotamia, The invention of the city| place= London| publisher= Penguin Books| year= 2002}}&lt;/ref&gt;.

In [[w:Archaeology|archaeology]], a '''[[w:Tell (archaeology)|tell]]''' is a mound consisting of the accumulated and stratified debris of a succession of consecutive settlements at the same site, the refuse of generations of people who built and inhabited them, alongside natural sediment. The most recent layers of a tell are nearest to the surface. As archaeologists dig into the tell, they encounter successively older layers of civilization, a recording of history called [[w:Stratigraphy|stratigraphy]]. 

The following table compares the stratigraphy found during the Eridu 1946–1949 campaigns with the stratigraphy found during the 1931 excavation of Shuruppak (Tell Fara). The table is organized according to the order in which layers are uncovered, with the youngest layers in the first rows of the table and successively older layers in subsequent rows. As shown below, the historic timeline separating the first construction at Eridu (c. 5300 BC) from the cultural break of the Shuruppak flood (c. 2900 BC) spans a real-world duration of approximately 2,400 years.

{| class=&quot;wikitable&quot;
! Historical Period &amp; Dating
! Eridu Temple Stratigraphy
! Shuruppak Stratigraphy
|-
| style=&quot;background-color:#ffdddd;&quot; | '''[[w:Sumu-abum|Amorite Era Begins (1894 BC)]]'''
| style=&quot;text-align: center;&quot; | –
| style=&quot;text-align: center;&quot; | –
| ''' Dynastic Periods '''
''(c. 1894-2900 BC)''
| '''Ur III Ziggurat Layer''' &lt;br/&gt; &quot;In the ruins of the ziggurat we found numerous bricks of&quot; [[w:Amar-Sin|Amar-Sin (c. 2046–2037 BC)]], &quot;three bricks of&quot; [[w:Ur-Nammu|Ur-Nammu (2112–2094 BC)]] &quot;one brick of&quot; [[w:Nur-Adad|Nur-Adad (c. 1866-1850 BC)]] &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;It seems that Ur-Nammu began the building of this ziggurat, but the bulk of the task was left to his grandson Amar-Sin to accomplish.&quot;]
| '''Fara II &amp; III Layer'''
([[w:Shu-Sin|Shu-Sin]] inscription (c. 2037 – 2028 BC), Sumerian seal cylinders, pottery, skeletons, figurines, Globular Stone Jar)
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 2900 BC'''
| style=&quot;text-align: center;&quot; | –
| style=&quot;background-color:#ffdddd;&quot; | '''The Shuruppak Flood Layer'''
[https://www.penn.museum/sites/journal/9356/ &quot;We found an alluvial layer at a depth between four and five metres below the mound surface.&quot;] [https://www.penn.museum/sites/journal/9356/ &quot;Only a few objects of this period, namely one stamp seal and some sherds “floated” in the refuse above. In this respect an absolute culture break took place.&quot;]
|-
| '''[[w:Jemdet Nasr period|Jemdet Nasr]]'''
''(c. 2900-3100 BC)''
| style=&quot;text-align: center;&quot; | –
| '''Fara I Layer''' 
(Scaraboid Stamp Seals, Polychrome Jar, Animal Patterns on Potsherds, Copper Blade)
|-
| '''[[w:Uruk period|Uruk]]'''
''(c. 3100-3800 BC)''
| '''Temple Layers I–V''' &lt;br/&gt; Temple I &amp; II: [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;All that is left to us are its foundations and retaining-walls, while even of these a large portion are buried beneath the massive masonry of the Third Dynasty Ziggurat.] &lt;br/&gt; Temple III: Small bricks of reddish clay. &lt;br/&gt; Temple IV: Medium sized bricks of greenish clay. &lt;br/&gt; Temple V: Large bricks of light-colored clay with dark joints. &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;Since Temple VI was now known to belong to a very late, if not the final stage of the Al-Ubaid period, Temples III, IV and V must be taken to represent a stage intermediate between this and the stone architecture of the Protoliterate age&quot;]
| style=&quot;text-align: center;&quot; | –
|-
| '''Ubaid 1-4'''
''(c. 3800-5300 BC)''
| '''Temples VI–XVIII''' &lt;br/&gt; Temple X: &quot;At Level X the temple appeared to have been rebuilt with only minor changes in the plan.&quot; &lt;br/&gt; Temple XI: &quot;provides the first instance of a temple building being deliberately raised upon a platform.&quot; &lt;br/&gt; Temple XII &amp; XIII: &quot;there were two successive occupation-levels at which no traces of actual buildings were to be found&quot;  &lt;br/&gt; Temple XVIII &quot;The first traces of occupation at Eridu occurred at a level corresponding to 11.70 meters beneath datum&quot; ... &quot;and, directly above, at Level XVII, were the remains of the earliest recognizable building&quot;
| style=&quot;text-align: center;&quot; | –
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 5300 BC'''
| style=&quot;background-color:#ffdddd;&quot; | '''First Construction'''
| style=&quot;text-align: center;&quot; | –
|}

=== Pre-flood Bible Chronology ===
'''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal. Writing from [https://commons.wikimedia.org/wiki/File:Ptolemaic_Alexandria_-_en.svg Alexandria in the late 3rd century BC (c. 221 BC)], Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood of Noah.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;11&quot; style=&quot;background-color:#e8e8e8;&quot; | —
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 230
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 205
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 190
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 170
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1309
|}

The Septuagint chronology, as witnessed by Demetrius at 2,264 years, mirrors this archaeological reality within a remarkably tight margin of error (less than 6%).
This broad-stroke alignment strongly indicates that while the fine internal details were arranged to fit sacred numerical cycles, the overarching framework of the Septuagint preserved an authentic, historically grounded memory of the Fertile Crescent's true antiquity—a memory that was severely contracted in alternative textual traditions.

The structural overlap between the Mesopotamian and biblical traditions becomes most compelling when evaluating the total duration of the antediluvian era. While the fine details and specific individual lifespans in both the Septuagint and the Sumerian King List are heavily calculated using stylized, sacred, or mathematically perfect numbers, the macro-span of the Septuagint timeline displays an uncanny alignment with real-world ancient history.

This macro-alignment is particularly remarkable when contrasted against the Sumerian King List itself. In the Mesopotamian scribal tradition, the antediluvian era spans astronomical lengths of time—typically totaling over 240,000 years across the reigns of the first eight pre-flood kings. These numbers are clearly symbolic or mathematical inflations that completely diverge from physical archaeological realities.

Rather than duplicating these vast Near Eastern timelines or severely compressing the era, the Septuagint tradition preserved a highly realistic historical interval. 

It is highly significant that the Septuagint chronology, as witnessed by Demetrius, aligns so closely with modern archaeological horizons instead of the exaggerated spans of the SKL. Because an unguided mathematical arrangement of sacred numbers would be statistically unlikely to land so close to actual physical history, this uncanny broad-stroke accuracy hints at two intriguing possibilities: it was either a case of monumental cosmic luck, or the Alexandrian redactors possessed ancient, pristine Near Eastern source records that have since been entirely lost to history.

This total of 2,264 years stands in stark contrast to the significantly compressed timelines preserved in the Masoretic Text (1,656 years) and the Samaritan Pentateuch (1,307 years). When evaluating these numbers against physical history, the Septuagint's macro-chronology reveals a striking real-world correlation.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].


==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167 / 187
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

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====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.


Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.
{{RoundBoxBottom}}
{{RoundBoxTop}}
'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
{{RoundBoxBottom}}

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors, as shown in the following tables.

(The '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the Black Desert of Jordan. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

{{RoundBoxTop}}
==== The Birth of Shem (A Digression) ====
Were Noah's sons born when Noah was 500 or 502?

While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.
{{RoundBoxBottom}}

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 - 1}{2} + 3^2 &amp;= 1200 + 9 \\
 &amp;= 1209
\end{aligned}
&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 + 1}{2} + (7^2 - 3^2) &amp;= 1201 + 40 \\
 &amp;= 1241
\end{aligned}
&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== The Masoretic text Variation ==


[[File:Schematic_Diagram_Masoretic_Text_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Masoretic Text'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

=== Mathematical Structure of the Masoretic Chronology ===
The above diagram maps the Masoretic chronology using the identical grid framework applied to the Samaritan Pentateuch. Red circles indicate specific chronological points where the Masoretic timeline diverges from the Samaritan tradition. Key textual and numerical variations include:

==== The Universal Flood Adjustment (+349 yrs) ====
The Masoretic chronology postpones the birth of Noah and the onset of the Deluge by '''349 years''' relative to the Samaritan tradition. As outlined in the above &quot;Masoretic Adjustments&quot; section, this structural shift ensures that Noah's antediluvian ancestors do not survive past the date of the universal flood. This timeline was achieved by adding 100 years to Jared's paternal age, 120 years to Methuselah's, and 129 years to Lamech's. Consequently, these calculations place Methuselah's death precisely in the year of the Flood and Lamech's death five years prior.

==== The Smoothing of Noah's Life (-2 yrs) ====
The Masoretic chronology reduces Noah's age at the birth of Shem from 502 years (as found in the Samaritan Pentateuch) to 500 years. As discussed in the &quot;Lectio Difficilior Potior&quot; and &quot;The Birth of Shem (A Digression)&quot; sections, this change represents &quot;scribal smoothing.&quot; This editorial process rounded major milestones in Noah's life to the nearest century: his sons are born in his 500th year, the Deluge occurs in his 600th year, and his father Lamech dies in his 700th year.

==== Post-Flood Patriarchal Lifespans (-650 yrs) ====
Seven post-flood patriarchs (Arpachshad through Nahor) exhibit reduced ages at the birth of their firstborn sons within the Masoretic chronology, resulting in a cumulative deficit of 650 years compared to the Samaritan Pentateuch. Specifically, the paternal ages of Arpachshad, Shelah, Eber, Peleg, Reu, and Serug are each lower by exactly 100 years, while Nahor's age is reduced by 50 years. Given the uniform, rounded values of these variations, a direct dependency exists; either the Masoretic chronologist systematically deflated these ages, or the Samaritan chronologist systematically inflated them.

The Masoretic use of young fathering ages for these specific patriarchs may preserve an older tradition. The narrative theme of Abraham fathering Isaac at the age of 100 serves as a central literary and mathematical motif that likely predates the Babylonian exile. Consequently, the lower paternal ages in the Masoretic lineage may intentionally highlight this milestone, ensuring that a fatherhood age of 100 remains an extraordinary, miraculous exception within the broader genealogical timeline.

Under this proposed framework, Arpachshad (or Cainan) originally functioned not as a standard descendant of Shem, but as the progenitor of a distinct nine-generation patriarchal line culminating in the miraculous birth of Isaac. During the Babylonian exile, the Priestly source (&quot;P&quot;) may have adapted this older Abrahamic lineage, appending earlier primeval generations to mirror Mesopotamian creation and flood epics. Consequently, Mesopotamian archetypes—such as Adapa, Enmeduranki, and Utnapishtim—were synthesized with preexisting Hebrew traditions to form the composite figures of Adam, Enoch, and Noah. These composite primeval figures were subsequently integrated into a single, seamless genealogical chain spanning from creation to the foundations of Israel.

==== The Final Masoretic Adjustment (+215 yrs) ====
As will be explained later in this resource, it is proposed that a final 215 years was added to the Masoretic chronology to arrive at the 2666-year total. The Samaritan Pentateuch, the Septuagint, and the Book of Jubilees all claim that the Patriarchs were in Canaan for 215 years, followed by 215 years of Israelite slavery in Egypt. The Masoretic text increases the period of Egyptian slavery by 215 years, for a total of 430 years of slavery.

This increase to 430 years is difficult to reconcile with the Moses narrative, which claims that Moses was the great-grandson of Levi on the paternal line and the grandson on the maternal line. Levi entered Egypt as an adult and a free man. Moses led the children of Israel out of Egypt at eighty years old. For Moses to be a maternal grandson, his mother would have been hundreds of years old when she gave birth to him and his older siblings, Aaron and Miriam.

==== The 2800-Year to 2666-Year Transition ====
Within the Masoretic chronology, the Exodus is situated inside the third 800-year epoch, occurring precisely in the year 2666 after creation (Anno Mundi). This placement represents a deliberate scribal design choice rather than an organic chronological coincidence.

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;

* '''Masoretic Text (Adam to Exodus):'''
:&lt;math display=&quot;block&quot;&gt;
(2800 - 46) + (349) - (2) - (650) + (215) = 2666&lt;/math&gt;
&lt;/div&gt;

As detailed in &quot;The 350-Year Symmetrical Extension,&quot; the timeline found in the Book of Jubilees likely served as the foundational model, positioning Shem at the exact midpoint of its 2,450-year chronology. The Samaritan Pentateuch appears to be a direct derivative of this Jubilees framework. Scribes systematically appended exactly 350 years to the post-Flood era, shifting the completion of the Conquest to a precise macro-structural target of 2,800 years after creation.

Envisioning the process that transitioned from the 2,800-year Samaritan chronology to the Masoretic framework presents a distinct mathematical challenge. Tracking the adjustments step-by-step from the Samaritan baseline reveals a precise transition:

* '''Samaritan Baseline (Conquest Completion):''' 2,800 years
* '''Minus Wandering and Conquest Era:''' −46 years (2,754)
* '''Plus Masoretic Flood Adjustment:''' +349 years (3,103)
* '''Minus Noah's Life Smoothing:''' −2 years (3,101)
* '''Minus Post-Flood Patriarchal Reductions:''' −650 years (2,451)
* '''The Final Masoretic Adjustment:''' +215 years (2,666)

This calculation yields exactly 2,666 years. Because individual components were dictated by rigid external theological and narrative constraints, it is surprising that the final adjustment yields exactly 2,666 years, but it also remains difficult to isolate where scribes might have executed fine-tuning to bring about this exact value and bridge the 2,800-year and 2,666-year macro-structures.

=== Mathematical Structure of the Septuagint Chronology ===
As was explained above in &quot;Septuagint Adjustments&quot;, the LXX’s editor methodically added 100 years to each pre-flood patriarch. The purpose of this adjustment was to ensure that Noah would be the exclusive survivor of the flood. As shown in the tables below, post-flood &quot;Long Chronology&quot; variations are seen to be either identical to or derived from &quot;Short Chronology&quot; values.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. The [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology shown above, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.

[[Category:Religion]]


=== Mythology ===
In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.</text>
      <sha1>1q95mq2e7m6ew8hlvmeqr6lb7ocmnah</sha1>
    </revision>
    <revision>
      <id>2845059</id>
      <parentid>2845058</parentid>
      <timestamp>2026-11-10T03:26:56Z</timestamp>
      <contributor>
        <username>CanonicalMormon</username>
        <id>2646631</id>
      </contributor>
      <comment>/* Chronological Similarities */</comment>
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      <text bytes="128195" sha1="0l0fumyo1h2vvaikxdhloxaluwih15p" xml:space="preserve">{| class=table style=&quot;width:100%;&quot;
|-
 | {{Original research}}
|-
 | {{RoundBoxTop}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional context and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. This article was researched and written by a human author, with editing assistance provided by [https://gemini.google.com Google &quot;Gemini, your personal AI assistant&quot;]{{RoundBoxBottom}}
|}

= The Sumerian Origins of Genesis =
The early history in Genesis shares deep structural, thematic, and chronological ties with ancient Mesopotamian writing traditions. There is a close parallel between the biblical figures listed in Genesis 5, who lived before the flood, and those listed in the '''[[w:Sumerian King List|Sumerian King List (SKL)]]'''. This ancient text details a succession of long-lived rulers who reigned right up until a cataclysmic flood destroyed their civilization. The table below shows the connections between Genesis 5 and the Sumerian King List, with directly matching details highlighted in blue.

{| class=&quot;nowraplinks&quot; cellspacing=&quot;0&quot; cellpadding=&quot;3&quot; rules=&quot;all&quot; style=&quot;background:#fbfbfb; border: 1px solid #ccc; font-size:90%; empty-cells:show; border-collapse:collapse; text-align:left;&quot;
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! colspan=&quot;2&quot; | Sumerian King List
! Hebrew Bible
|- style=&quot;text-align:center; font-style:italic;&quot;
! colspan=&quot;2&quot; style=&quot;background:#fffcf2;&quot; | Archaeological Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2400&lt;/span&gt; Years Total
! style=&quot;background:#fffcf2;&quot; | Demetrius Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2264&lt;/span&gt; Years Total
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! City
! Ruler (Sage)
! Patriarch
|-
| [[w:Eridu|Eridu]]
| [[w:Alulim|Alulim]] &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;([[w:Adapa|Adapa]])&lt;/span&gt; &lt;br /&gt; [[w:Alalngar|Alalngar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Adam|Adam]]&lt;/span&gt; &lt;br /&gt; [[w:Seth|Seth]]
|-
| [[w:Bad-tibira|Bad-tibira]]
| [[w:En-men-lu-ana|En-men-lu-ana]] &lt;br /&gt; [[w:En-men-gal-ana|En-men-gal-ana]] &lt;br /&gt; [[w:Dumuzid|Dumuzid]] 
| [[w:Enos|Enos]] &lt;br /&gt; [[w:Kenan|Kenan]] &lt;br /&gt; [[w:Mahalalel|Mahalalel]]
|-
| [[w:Larak (Sumer)|Larak]] 
| [[w:En-sipad-zid-ana|En-sipad-zid-ana]] 
| [[w:Jared (biblical figure)|Jared]]
|-
| [[w:Sippar|Sippar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:En-men-dur-ana|En-men-dur-ana]]&lt;/span&gt; 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Enoch|Enoch]]&lt;/span&gt;
|-

| [[w:Shuruppak|Shuruppak]]
| [[w:Ubara-Tutu|Ubara-Tutu]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Utnapishtim|Utnapishtim]]&lt;/span&gt; 
| [[w:Methuselah|Methuselah]] &lt;br /&gt; [[w:Lamech (father of Noah)|Lamech]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Noah|Noah]]&lt;/span&gt;
|}

== Narrative Similarities ==
The biblical accounts seem to build on, or develop alongside, the legendary roles given to Sumerian rulers and their wise advisors, known as apkallu (sages). Key thematic connections include:

*[https://www.worldhistory.org/article/216/the-myth-of-adapa/ Adam as Adapa] : In the Adapa narrative, Ea (god of wisdom) warns Adapa to refuse any food or drink offered in heaven but to accept clothing and oil. Anu (the sky god) offers Adapa food and drink, followed by clothing and oil. When Adapa refuses the food, Anu chides him for missing an opportunity for immortality. Similarly, in the Genesis account, Yahweh Elohim permits Adam and Eve to eat from any tree in the divine garden except the tree of knowledge. A competing figure, the serpent, advises them to eat from it anyway. After they do, Yahweh returns, chastises them for disobedience, and provides garments of skin. Key parallels between the accounts include: A potential similarity and shared root in the names Adapa and Adam. Both narratives feature a test involving the consumption of supposedly deadly food. Both figures are offered clothing and summoned before a deity to answer for their decisions.

*[https://www.marquette.edu/maqom/Chapter%201.pdf Enoch as Enmeduranki]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew Book of Enoch describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text Pirišti Šamê u Erṣeti (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.

*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

== Chronological Similarities ==
[[File:Antediluvian Sumerian King List Cities and Biblical Rivers.png|thumb|400px|Locations of four of the five pre-Flood cities named in the Sumerian King List have been identified through excavated cuneiform inscriptions. These cities existed in close proximity to the Tigris and Euphrates rivers. Eridu was a coastal city when first founded around 5300 BC. Over the millennia, however, massive deposits of alluvial silt from four river systems filled the ancient bays, gradually pushing the shoreline southeastward away from Eridu. There are four rivers named in the biblical Garden of Eden story (Genesis 2:10–14): the Tigris, Euphrates, Gihon, and Pishon. The Gihon and Pishon are frequently associated with the Karun River in western Iran and the Wadi al-Batin (a now-dry riverbed) in the Arabian Peninsula. The blue oval on the map above represents the 5300 BC coastline, with arrows pointing to the mouths of the four Eden rivers.]]

In Mesopotamian mythology, Eridu was the oldest city, the home of the wisdom god Enki, and the first site where divine kingship descended. While many Sumerian texts omit the pre-Flood era entirely, in narratives that do include a pre-Flood prologue, Eridu is consistently identified as the first city and Šuruppak is frequently identified as the fifth and final city prior to the Flood.

One English translation of the Sumerian King List states it as follows: [https://www.livius.org/sources/content/anet/266-the-sumerian-king-list/ &quot;After the kingship descended from heaven, the kingship was in Eridu ... Bad-tibira fell and the kingship was taken to Larak ... Sippar fell and the kingship was taken to Šuruppak ... Five cities; eight kings ruled for 385,200 years. Then the Flood swept over.&quot;]

A related text, the [[w:Eridu Genesis|Eridu Genesis]], preserves a parallel tradition with Eridu as the first city and Šuruppak as the fifth: [https://www.livius.org/sources/content/oriental-varia/eridu-genesis/ &quot;The firstling of the cities, Eridu, she gave to the leader Nudimmud, the second, Bad-Tibira, she gave to the Prince and the Sacred One, the third, Larak, she gave to Pahilsag, the fourth, Sippar, she gave to the gallant Utu, the fifth, Šuruppak, she gave to Ansud.&quot;]

Locations of four of the five pre-Flood Sumerian cities have been identified archeologically. Eridu sits on a hill inside a basin roughly {{convert|15|mi}} long and {{convert|20|ft}} deep, separated from the Euphrates by a sandstone ridge called the ''Hazem''. In the 3rd Millennium BC, before the river shifted course, a canal connected Eridu to the Euphrates. Low tells with 2nd Millennium BC pottery and later burials mark the canal's path.

====Prehistoric Temple(s)====
[[File:Ea (Babilonian) - EnKi (Sumerian).jpg|thumb|Detail of the Adda Seal, an ancient Akkadian cylinder seal (circa 2300 BCE) depicting Enki with water streams coming from his shoulders.|270x270px]]
The urban nucleus of Eridu was Enki's temple, called the ''House of the Aquifer'' in the oldest Sumerian texts. In later texts, this temple was called the ''House of the Waters''. [[w:Enki|Enki]] was the Mesopotamian god of subterranean waters, specifically reigning over the primeval underground ocean known as the [[w:Abzu|Abzu]]. In ancient art, he was frequently represented with water streams flowing directly from his body, which were often filled with fish swimming through the currents. The biblical [[w:Garden of Eden|Garden of Eden]] is presented as a well-watered garden with rivers running out of it. The Sumerian association of Eridu with subterranean waters seems reflected in the biblical garden story.

[[File:Eridu Level Sequence A.jpg|frame|Prehistoric sequence of buildings excavated at Eridu.]]
A prehistoric sequence of buildings excavated at Eridu during the 1946 to 1949 campaigns seems to substantiate Eridu's status as the first city with the earliest temple. Consisting of eighteen occupational levels, this Eridu sequence of &quot;Palace-like&quot; or &quot;Temple-Like&quot; buildings appears to be the first occurrence of such structures in the region&lt;ref&gt;{{cite book| last= Leick| first= Gwendolyn| title= Mesopotamia, The invention of the city| place= London| publisher= Penguin Books| year= 2002}}&lt;/ref&gt;.

In [[w:Archaeology|archaeology]], a '''[[w:Tell (archaeology)|tell]]''' is a mound consisting of the accumulated and stratified debris of a succession of consecutive settlements at the same site, the refuse of generations of people who built and inhabited them, alongside natural sediment. The most recent layers of a tell are nearest to the surface. As archaeologists dig into the tell, they encounter successively older layers of civilization, a recording of history called [[w:Stratigraphy|stratigraphy]]. 

The following table compares the stratigraphy found during the Eridu 1946–1949 campaigns with the stratigraphy found during the 1931 excavation of Shuruppak (Tell Fara). The table is organized according to the order in which layers are uncovered, with the youngest layers in the first rows of the table and successively older layers in subsequent rows. As shown below, the historic timeline separating the first construction at Eridu (c. 5300 BC) from the cultural break of the Shuruppak flood (c. 2900 BC) spans a real-world duration of approximately 2,400 years.

{| class=&quot;wikitable&quot;
! Historical Period &amp; Dating
! Eridu Temple Stratigraphy
! Shuruppak Stratigraphy
|-
| style=&quot;background-color:#ffdddd;&quot; | '''[[w:Sumu-abum|Amorite Era Begins (1894 BC)]]'''
| style=&quot;text-align: center;&quot; | –
| style=&quot;text-align: center;&quot; | –
|-
| ''' Dynastic Periods '''
''(c. 1894-2900 BC)''
| '''Ur III Ziggurat Layer''' &lt;br/&gt; &quot;In the ruins of the ziggurat we found numerous bricks of&quot; [[w:Amar-Sin|Amar-Sin (c. 2046–2037 BC)]], &quot;three bricks of&quot; [[w:Ur-Nammu|Ur-Nammu (2112–2094 BC)]] &quot;one brick of&quot; [[w:Nur-Adad|Nur-Adad (c. 1866-1850 BC)]] &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;It seems that Ur-Nammu began the building of this ziggurat, but the bulk of the task was left to his grandson Amar-Sin to accomplish.&quot;]
| '''Fara II &amp; III Layer'''
([[w:Shu-Sin|Shu-Sin]] inscription (c. 2037 – 2028 BC), Sumerian seal cylinders, pottery, skeletons, figurines, Globular Stone Jar)
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 2900 BC'''
| style=&quot;text-align: center;&quot; | –
| style=&quot;background-color:#ffdddd;&quot; | '''The Shuruppak Flood Layer'''
[https://www.penn.museum/sites/journal/9356/ &quot;We found an alluvial layer at a depth between four and five metres below the mound surface.&quot;] [https://www.penn.museum/sites/journal/9356/ &quot;Only a few objects of this period, namely one stamp seal and some sherds “floated” in the refuse above. In this respect an absolute culture break took place.&quot;]
|-
| '''[[w:Jemdet Nasr period|Jemdet Nasr]]'''
''(c. 2900-3100 BC)''
| style=&quot;text-align: center;&quot; | –
| '''Fara I Layer''' 
(Scaraboid Stamp Seals, Polychrome Jar, Animal Patterns on Potsherds, Copper Blade)
|-
| '''[[w:Uruk period|Uruk]]'''
''(c. 3100-3800 BC)''
| '''Temple Layers I–V''' &lt;br/&gt; Temple I &amp; II: [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;All that is left to us are its foundations and retaining-walls, while even of these a large portion are buried beneath the massive masonry of the Third Dynasty Ziggurat.] &lt;br/&gt; Temple III: Small bricks of reddish clay. &lt;br/&gt; Temple IV: Medium sized bricks of greenish clay. &lt;br/&gt; Temple V: Large bricks of light-colored clay with dark joints. &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;Since Temple VI was now known to belong to a very late, if not the final stage of the Al-Ubaid period, Temples III, IV and V must be taken to represent a stage intermediate between this and the stone architecture of the Protoliterate age&quot;]
| style=&quot;text-align: center;&quot; | –
|-
| '''Ubaid 1-4'''
''(c. 3800-5300 BC)''
| '''Temples VI–XVIII''' &lt;br/&gt; Temple X: &quot;At Level X the temple appeared to have been rebuilt with only minor changes in the plan.&quot; &lt;br/&gt; Temple XI: &quot;provides the first instance of a temple building being deliberately raised upon a platform.&quot; &lt;br/&gt; Temple XII &amp; XIII: &quot;there were two successive occupation-levels at which no traces of actual buildings were to be found&quot;  &lt;br/&gt; Temple XVIII &quot;The first traces of occupation at Eridu occurred at a level corresponding to 11.70 meters beneath datum&quot; ... &quot;and, directly above, at Level XVII, were the remains of the earliest recognizable building&quot;
| style=&quot;text-align: center;&quot; | –
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 5300 BC'''
| style=&quot;background-color:#ffdddd;&quot; | '''First Construction'''
| style=&quot;text-align: center;&quot; | –
|}

=== Pre-flood Bible Chronology ===
'''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal. Writing from [https://commons.wikimedia.org/wiki/File:Ptolemaic_Alexandria_-_en.svg Alexandria in the late 3rd century BC (c. 221 BC)], Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood of Noah.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;11&quot; style=&quot;background-color:#e8e8e8;&quot; | —
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 230
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 205
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 190
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 170
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1309
|}

The Septuagint chronology, as witnessed by Demetrius at 2,264 years, mirrors this archaeological reality within a remarkably tight margin of error (less than 6%).
This broad-stroke alignment strongly indicates that while the fine internal details were arranged to fit sacred numerical cycles, the overarching framework of the Septuagint preserved an authentic, historically grounded memory of the Fertile Crescent's true antiquity—a memory that was severely contracted in alternative textual traditions.

The structural overlap between the Mesopotamian and biblical traditions becomes most compelling when evaluating the total duration of the antediluvian era. While the fine details and specific individual lifespans in both the Septuagint and the Sumerian King List are heavily calculated using stylized, sacred, or mathematically perfect numbers, the macro-span of the Septuagint timeline displays an uncanny alignment with real-world ancient history.

This macro-alignment is particularly remarkable when contrasted against the Sumerian King List itself. In the Mesopotamian scribal tradition, the antediluvian era spans astronomical lengths of time—typically totaling over 240,000 years across the reigns of the first eight pre-flood kings. These numbers are clearly symbolic or mathematical inflations that completely diverge from physical archaeological realities.

Rather than duplicating these vast Near Eastern timelines or severely compressing the era, the Septuagint tradition preserved a highly realistic historical interval. 

It is highly significant that the Septuagint chronology, as witnessed by Demetrius, aligns so closely with modern archaeological horizons instead of the exaggerated spans of the SKL. Because an unguided mathematical arrangement of sacred numbers would be statistically unlikely to land so close to actual physical history, this uncanny broad-stroke accuracy hints at two intriguing possibilities: it was either a case of monumental cosmic luck, or the Alexandrian redactors possessed ancient, pristine Near Eastern source records that have since been entirely lost to history.

This total of 2,264 years stands in stark contrast to the significantly compressed timelines preserved in the Masoretic Text (1,656 years) and the Samaritan Pentateuch (1,307 years). When evaluating these numbers against physical history, the Septuagint's macro-chronology reveals a striking real-world correlation.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].


==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167 / 187
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

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====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.


Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.
{{RoundBoxBottom}}
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'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
{{RoundBoxBottom}}

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors, as shown in the following tables.

(The '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the Black Desert of Jordan. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

{{RoundBoxTop}}
==== The Birth of Shem (A Digression) ====
Were Noah's sons born when Noah was 500 or 502?

While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.
{{RoundBoxBottom}}

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 - 1}{2} + 3^2 &amp;= 1200 + 9 \\
 &amp;= 1209
\end{aligned}
&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 + 1}{2} + (7^2 - 3^2) &amp;= 1201 + 40 \\
 &amp;= 1241
\end{aligned}
&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== The Masoretic text Variation ==


[[File:Schematic_Diagram_Masoretic_Text_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Masoretic Text'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

=== Mathematical Structure of the Masoretic Chronology ===
The above diagram maps the Masoretic chronology using the identical grid framework applied to the Samaritan Pentateuch. Red circles indicate specific chronological points where the Masoretic timeline diverges from the Samaritan tradition. Key textual and numerical variations include:

==== The Universal Flood Adjustment (+349 yrs) ====
The Masoretic chronology postpones the birth of Noah and the onset of the Deluge by '''349 years''' relative to the Samaritan tradition. As outlined in the above &quot;Masoretic Adjustments&quot; section, this structural shift ensures that Noah's antediluvian ancestors do not survive past the date of the universal flood. This timeline was achieved by adding 100 years to Jared's paternal age, 120 years to Methuselah's, and 129 years to Lamech's. Consequently, these calculations place Methuselah's death precisely in the year of the Flood and Lamech's death five years prior.

==== The Smoothing of Noah's Life (-2 yrs) ====
The Masoretic chronology reduces Noah's age at the birth of Shem from 502 years (as found in the Samaritan Pentateuch) to 500 years. As discussed in the &quot;Lectio Difficilior Potior&quot; and &quot;The Birth of Shem (A Digression)&quot; sections, this change represents &quot;scribal smoothing.&quot; This editorial process rounded major milestones in Noah's life to the nearest century: his sons are born in his 500th year, the Deluge occurs in his 600th year, and his father Lamech dies in his 700th year.

==== Post-Flood Patriarchal Lifespans (-650 yrs) ====
Seven post-flood patriarchs (Arpachshad through Nahor) exhibit reduced ages at the birth of their firstborn sons within the Masoretic chronology, resulting in a cumulative deficit of 650 years compared to the Samaritan Pentateuch. Specifically, the paternal ages of Arpachshad, Shelah, Eber, Peleg, Reu, and Serug are each lower by exactly 100 years, while Nahor's age is reduced by 50 years. Given the uniform, rounded values of these variations, a direct dependency exists; either the Masoretic chronologist systematically deflated these ages, or the Samaritan chronologist systematically inflated them.

The Masoretic use of young fathering ages for these specific patriarchs may preserve an older tradition. The narrative theme of Abraham fathering Isaac at the age of 100 serves as a central literary and mathematical motif that likely predates the Babylonian exile. Consequently, the lower paternal ages in the Masoretic lineage may intentionally highlight this milestone, ensuring that a fatherhood age of 100 remains an extraordinary, miraculous exception within the broader genealogical timeline.

Under this proposed framework, Arpachshad (or Cainan) originally functioned not as a standard descendant of Shem, but as the progenitor of a distinct nine-generation patriarchal line culminating in the miraculous birth of Isaac. During the Babylonian exile, the Priestly source (&quot;P&quot;) may have adapted this older Abrahamic lineage, appending earlier primeval generations to mirror Mesopotamian creation and flood epics. Consequently, Mesopotamian archetypes—such as Adapa, Enmeduranki, and Utnapishtim—were synthesized with preexisting Hebrew traditions to form the composite figures of Adam, Enoch, and Noah. These composite primeval figures were subsequently integrated into a single, seamless genealogical chain spanning from creation to the foundations of Israel.

==== The Final Masoretic Adjustment (+215 yrs) ====
As will be explained later in this resource, it is proposed that a final 215 years was added to the Masoretic chronology to arrive at the 2666-year total. The Samaritan Pentateuch, the Septuagint, and the Book of Jubilees all claim that the Patriarchs were in Canaan for 215 years, followed by 215 years of Israelite slavery in Egypt. The Masoretic text increases the period of Egyptian slavery by 215 years, for a total of 430 years of slavery.

This increase to 430 years is difficult to reconcile with the Moses narrative, which claims that Moses was the great-grandson of Levi on the paternal line and the grandson on the maternal line. Levi entered Egypt as an adult and a free man. Moses led the children of Israel out of Egypt at eighty years old. For Moses to be a maternal grandson, his mother would have been hundreds of years old when she gave birth to him and his older siblings, Aaron and Miriam.

==== The 2800-Year to 2666-Year Transition ====
Within the Masoretic chronology, the Exodus is situated inside the third 800-year epoch, occurring precisely in the year 2666 after creation (Anno Mundi). This placement represents a deliberate scribal design choice rather than an organic chronological coincidence.

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;

* '''Masoretic Text (Adam to Exodus):'''
:&lt;math display=&quot;block&quot;&gt;
(2800 - 46) + (349) - (2) - (650) + (215) = 2666&lt;/math&gt;
&lt;/div&gt;

As detailed in &quot;The 350-Year Symmetrical Extension,&quot; the timeline found in the Book of Jubilees likely served as the foundational model, positioning Shem at the exact midpoint of its 2,450-year chronology. The Samaritan Pentateuch appears to be a direct derivative of this Jubilees framework. Scribes systematically appended exactly 350 years to the post-Flood era, shifting the completion of the Conquest to a precise macro-structural target of 2,800 years after creation.

Envisioning the process that transitioned from the 2,800-year Samaritan chronology to the Masoretic framework presents a distinct mathematical challenge. Tracking the adjustments step-by-step from the Samaritan baseline reveals a precise transition:

* '''Samaritan Baseline (Conquest Completion):''' 2,800 years
* '''Minus Wandering and Conquest Era:''' −46 years (2,754)
* '''Plus Masoretic Flood Adjustment:''' +349 years (3,103)
* '''Minus Noah's Life Smoothing:''' −2 years (3,101)
* '''Minus Post-Flood Patriarchal Reductions:''' −650 years (2,451)
* '''The Final Masoretic Adjustment:''' +215 years (2,666)

This calculation yields exactly 2,666 years. Because individual components were dictated by rigid external theological and narrative constraints, it is surprising that the final adjustment yields exactly 2,666 years, but it also remains difficult to isolate where scribes might have executed fine-tuning to bring about this exact value and bridge the 2,800-year and 2,666-year macro-structures.

=== Mathematical Structure of the Septuagint Chronology ===
As was explained above in &quot;Septuagint Adjustments&quot;, the LXX’s editor methodically added 100 years to each pre-flood patriarch. The purpose of this adjustment was to ensure that Noah would be the exclusive survivor of the flood. As shown in the tables below, post-flood &quot;Long Chronology&quot; variations are seen to be either identical to or derived from &quot;Short Chronology&quot; values.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. The [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology shown above, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.

[[Category:Religion]]


=== Mythology ===
In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.</text>
      <sha1>0l0fumyo1h2vvaikxdhloxaluwih15p</sha1>
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    <revision>
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      <text bytes="128295" sha1="5lidsyi1ybvfl91qm68e1prirui6oim" xml:space="preserve">{| class=table style=&quot;width:100%;&quot;
|-
 | {{Original research}}
|-
 | {{RoundBoxTop}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional context and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. This article was researched and written by a human author, with editing assistance provided by [https://gemini.google.com Google &quot;Gemini, your personal AI assistant&quot;]{{RoundBoxBottom}}
|}

= The Sumerian Origins of Genesis =
The early history in Genesis shares deep structural, thematic, and chronological ties with ancient Mesopotamian writing traditions. There is a close parallel between the biblical figures listed in Genesis 5, who lived before the flood, and those listed in the '''[[w:Sumerian King List|Sumerian King List (SKL)]]'''. This ancient text details a succession of long-lived rulers who reigned right up until a cataclysmic flood destroyed their civilization. The table below shows the connections between Genesis 5 and the Sumerian King List, with directly matching details highlighted in blue.

{| class=&quot;nowraplinks&quot; cellspacing=&quot;0&quot; cellpadding=&quot;3&quot; rules=&quot;all&quot; style=&quot;background:#fbfbfb; border: 1px solid #ccc; font-size:90%; empty-cells:show; border-collapse:collapse; text-align:left;&quot;
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! colspan=&quot;2&quot; | Sumerian King List
! Hebrew Bible
|- style=&quot;text-align:center; font-style:italic;&quot;
! colspan=&quot;2&quot; style=&quot;background:#fffcf2;&quot; | Archaeological Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2400&lt;/span&gt; Years Total
! style=&quot;background:#fffcf2;&quot; | Demetrius Timeline:&lt;br&gt;&lt;span style=&quot;background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;&quot;&gt;2264&lt;/span&gt; Years Total
|- style=&quot;background:#F6E6AE; font-weight:bold; text-align:center;&quot;
! City
! Ruler (Sage)
! Patriarch
|-
| [[w:Eridu|Eridu]]
| [[w:Alulim|Alulim]] &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;([[w:Adapa|Adapa]])&lt;/span&gt; &lt;br /&gt; [[w:Alalngar|Alalngar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Adam|Adam]]&lt;/span&gt; &lt;br /&gt; [[w:Seth|Seth]]
|-
| [[w:Bad-tibira|Bad-tibira]]
| [[w:En-men-lu-ana|En-men-lu-ana]] &lt;br /&gt; [[w:En-men-gal-ana|En-men-gal-ana]] &lt;br /&gt; [[w:Dumuzid|Dumuzid]] 
| [[w:Enos|Enos]] &lt;br /&gt; [[w:Kenan|Kenan]] &lt;br /&gt; [[w:Mahalalel|Mahalalel]]
|-
| [[w:Larak (Sumer)|Larak]] 
| [[w:En-sipad-zid-ana|En-sipad-zid-ana]] 
| [[w:Jared (biblical figure)|Jared]]
|-
| [[w:Sippar|Sippar]] 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:En-men-dur-ana|En-men-dur-ana]]&lt;/span&gt; 
| &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Enoch|Enoch]]&lt;/span&gt;
|-

| [[w:Shuruppak|Shuruppak]]
| [[w:Ubara-Tutu|Ubara-Tutu]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Utnapishtim|Utnapishtim]]&lt;/span&gt; 
| [[w:Methuselah|Methuselah]] &lt;br /&gt; [[w:Lamech (father of Noah)|Lamech]] &lt;br /&gt; &lt;span style=&quot;background: #d6e2ff; padding: 0 2px;&quot;&gt;[[w:Noah|Noah]]&lt;/span&gt;
|}

== Narrative Similarities ==
The biblical accounts seem to build on, or develop alongside, the legendary roles given to Sumerian rulers and their wise advisors, known as apkallu (sages). Key thematic connections include:

*[https://www.worldhistory.org/article/216/the-myth-of-adapa/ Adam as Adapa] : In the Adapa narrative, Ea (god of wisdom) warns Adapa to refuse any food or drink offered in heaven but to accept clothing and oil. Anu (the sky god) offers Adapa food and drink, followed by clothing and oil. When Adapa refuses the food, Anu chides him for missing an opportunity for immortality. Similarly, in the Genesis account, Yahweh Elohim permits Adam and Eve to eat from any tree in the divine garden except the tree of knowledge. A competing figure, the serpent, advises them to eat from it anyway. After they do, Yahweh returns, chastises them for disobedience, and provides garments of skin. Key parallels between the accounts include: A potential similarity and shared root in the names Adapa and Adam. Both narratives feature a test involving the consumption of supposedly deadly food. Both figures are offered clothing and summoned before a deity to answer for their decisions.

*[https://www.marquette.edu/maqom/Chapter%201.pdf Enoch as Enmeduranki]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew Book of Enoch describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text Pirišti Šamê u Erṣeti (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.

*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

== Chronological Similarities ==
[[File:Antediluvian Sumerian King List Cities and Biblical Rivers.png|thumb|400px|Locations of four of the five pre-Flood cities named in the Sumerian King List have been identified through excavated cuneiform inscriptions. These cities existed in close proximity to the Tigris and Euphrates rivers. Eridu was a coastal city when first founded around 5300 BC. Over the millennia, however, massive deposits of alluvial silt from four river systems filled the ancient bays, gradually pushing the shoreline southeastward away from Eridu. There are four rivers named in the biblical Garden of Eden story (Genesis 2:10–14): the Tigris, Euphrates, Gihon, and Pishon. The Gihon and Pishon are frequently associated with the Karun River in western Iran and the Wadi al-Batin (a now-dry riverbed) in the Arabian Peninsula. The blue oval on the map above represents the 5300 BC coastline, with arrows pointing to the mouths of the four Eden rivers.]]

In Mesopotamian mythology, Eridu was the oldest city, the home of the wisdom god Enki, and the first site where divine kingship descended. While many Sumerian texts omit the pre-Flood era entirely, in narratives that do include a pre-Flood prologue, Eridu is consistently identified as the first city and Šuruppak is frequently identified as the fifth and final city prior to the Flood.

One English translation of the Sumerian King List states it as follows: [https://www.livius.org/sources/content/anet/266-the-sumerian-king-list/ &quot;After the kingship descended from heaven, the kingship was in Eridu ... Bad-tibira fell and the kingship was taken to Larak ... Sippar fell and the kingship was taken to Šuruppak ... Five cities; eight kings ruled for 385,200 years. Then the Flood swept over.&quot;]

A related text, the [[w:Eridu Genesis|Eridu Genesis]], preserves a parallel tradition with Eridu as the first city and Šuruppak as the fifth: [https://www.livius.org/sources/content/oriental-varia/eridu-genesis/ &quot;The firstling of the cities, Eridu, she gave to the leader Nudimmud, the second, Bad-Tibira, she gave to the Prince and the Sacred One, the third, Larak, she gave to Pahilsag, the fourth, Sippar, she gave to the gallant Utu, the fifth, Šuruppak, she gave to Ansud.&quot;]

Locations of four of the five pre-Flood Sumerian cities have been identified archeologically. Eridu sits on a hill inside a basin roughly {{convert|15|mi}} long and {{convert|20|ft}} deep, separated from the Euphrates by a sandstone ridge called the ''Hazem''. In the 3rd Millennium BC, before the river shifted course, a canal connected Eridu to the Euphrates. Low tells with 2nd Millennium BC pottery and later burials mark the canal's path.

====Prehistoric Temple(s)====
[[File:Ea (Babilonian) - EnKi (Sumerian).jpg|thumb|Detail of the Adda Seal, an ancient Akkadian cylinder seal (circa 2300 BCE) depicting Enki with water streams coming from his shoulders.|270x270px]]
The urban nucleus of Eridu was Enki's temple, called the ''House of the Aquifer'' in the oldest Sumerian texts. In later texts, this temple was called the ''House of the Waters''. [[w:Enki|Enki]] was the Mesopotamian god of subterranean waters, specifically reigning over the primeval underground ocean known as the [[w:Abzu|Abzu]]. In ancient art, he was frequently represented with water streams flowing directly from his body, which were often filled with fish swimming through the currents. The biblical [[w:Garden of Eden|Garden of Eden]] is presented as a well-watered garden with rivers running out of it. The Sumerian association of Eridu with subterranean waters seems reflected in the biblical garden story.

[[File:Eridu Level Sequence A.jpg|frame|Prehistoric sequence of buildings excavated at Eridu.]]
A prehistoric sequence of buildings excavated at Eridu during the 1946 to 1949 campaigns seems to substantiate Eridu's status as the first city with the earliest temple. Consisting of eighteen occupational levels, this Eridu sequence of &quot;Palace-like&quot; or &quot;Temple-Like&quot; buildings appears to be the first occurrence of such structures in the region&lt;ref&gt;{{cite book| last= Leick| first= Gwendolyn| title= Mesopotamia, The invention of the city| place= London| publisher= Penguin Books| year= 2002}}&lt;/ref&gt;.

In [[w:Archaeology|archaeology]], a '''[[w:Tell (archaeology)|tell]]''' is a mound consisting of the accumulated and stratified debris of a succession of consecutive settlements at the same site, the refuse of generations of people who built and inhabited them, alongside natural sediment. The most recent layers of a tell are nearest to the surface. As archaeologists dig into the tell, they encounter successively older layers of civilization, a recording of history called [[w:Stratigraphy|stratigraphy]]. 

The following table compares the stratigraphy found during the Eridu 1946–1949 campaigns with the stratigraphy found during the 1931 excavation of Shuruppak (Tell Fara). The table is organized according to the order in which layers are uncovered, with the youngest layers in the first rows of the table and successively older layers in subsequent rows. As shown below, the historic timeline separating the first construction at Eridu (c. 5300 BC) from the cultural break of the Shuruppak flood (c. 2900 BC) spans a real-world duration of approximately 2,400 years.

{| class=&quot;wikitable&quot;
! Historical Period &amp; Dating
! Eridu Temple Stratigraphy
! Shuruppak Stratigraphy
|-
| style=&quot;background-color:#ffdddd;&quot; | '''[[w:Sumu-abum|Amorite Era Begins (1894 BC)]]'''
| style=&quot;text-align: center;background-color:#ffdddd;&quot; | –
| style=&quot;text-align: center;background-color:#ffdddd;&quot; | –
|-
| ''' Dynastic Periods '''
''(c. 1894-2900 BC)''
| '''Ur III Ziggurat Layer''' &lt;br/&gt; &quot;In the ruins of the ziggurat we found numerous bricks of&quot; [[w:Amar-Sin|Amar-Sin (c. 2046–2037 BC)]], &quot;three bricks of&quot; [[w:Ur-Nammu|Ur-Nammu (2112–2094 BC)]] &quot;one brick of&quot; [[w:Nur-Adad|Nur-Adad (c. 1866-1850 BC)]] &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;It seems that Ur-Nammu began the building of this ziggurat, but the bulk of the task was left to his grandson Amar-Sin to accomplish.&quot;]
| '''Fara II &amp; III Layer'''
([[w:Shu-Sin|Shu-Sin]] inscription (c. 2037 – 2028 BC), Sumerian seal cylinders, pottery, skeletons, figurines, Globular Stone Jar)
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 2900 BC'''
| style=&quot;text-align: center;background-color:#ffdddd;&quot; | –
| style=&quot;background-color:#ffdddd;&quot; | '''The Shuruppak Flood Layer'''
[https://www.penn.museum/sites/journal/9356/ &quot;We found an alluvial layer at a depth between four and five metres below the mound surface.&quot;] [https://www.penn.museum/sites/journal/9356/ &quot;Only a few objects of this period, namely one stamp seal and some sherds “floated” in the refuse above. In this respect an absolute culture break took place.&quot;]
|-
| '''[[w:Jemdet Nasr period|Jemdet Nasr]]'''
''(c. 2900-3100 BC)''
| style=&quot;text-align: center;&quot; | –
| '''Fara I Layer''' 
(Scaraboid Stamp Seals, Polychrome Jar, Animal Patterns on Potsherds, Copper Blade)
|-
| '''[[w:Uruk period|Uruk]]'''
''(c. 3100-3800 BC)''
| '''Temple Layers I–V''' &lt;br/&gt; Temple I &amp; II: [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;All that is left to us are its foundations and retaining-walls, while even of these a large portion are buried beneath the massive masonry of the Third Dynasty Ziggurat.] &lt;br/&gt; Temple III: Small bricks of reddish clay. &lt;br/&gt; Temple IV: Medium sized bricks of greenish clay. &lt;br/&gt; Temple V: Large bricks of light-colored clay with dark joints. &lt;br/&gt; [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd &quot;Since Temple VI was now known to belong to a very late, if not the final stage of the Al-Ubaid period, Temples III, IV and V must be taken to represent a stage intermediate between this and the stone architecture of the Protoliterate age&quot;]
| style=&quot;text-align: center;&quot; | –
|-
| '''Ubaid 1-4'''
''(c. 3800-5300 BC)''
| '''Temples VI–XVIII''' &lt;br/&gt; Temple X: &quot;At Level X the temple appeared to have been rebuilt with only minor changes in the plan.&quot; &lt;br/&gt; Temple XI: &quot;provides the first instance of a temple building being deliberately raised upon a platform.&quot; &lt;br/&gt; Temple XII &amp; XIII: &quot;there were two successive occupation-levels at which no traces of actual buildings were to be found&quot;  &lt;br/&gt; Temple XVIII &quot;The first traces of occupation at Eridu occurred at a level corresponding to 11.70 meters beneath datum&quot; ... &quot;and, directly above, at Level XVII, were the remains of the earliest recognizable building&quot;
| style=&quot;text-align: center;&quot; | –
|-
| style=&quot;background-color:#ffdddd;&quot; | '''c. 5300 BC'''
| style=&quot;background-color:#ffdddd;&quot; | '''First Construction'''
| style=&quot;text-align: center;background-color:#ffdddd;&quot; | –
|}

=== Pre-flood Bible Chronology ===
'''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal. Writing from [https://commons.wikimedia.org/wiki/File:Ptolemaic_Alexandria_-_en.svg Alexandria in the late 3rd century BC (c. 221 BC)], Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood of Noah.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;11&quot; style=&quot;background-color:#e8e8e8;&quot; | —
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 230
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 205
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 190
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 170
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 165
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;4&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1309
|}

The Septuagint chronology, as witnessed by Demetrius at 2,264 years, mirrors this archaeological reality within a remarkably tight margin of error (less than 6%).
This broad-stroke alignment strongly indicates that while the fine internal details were arranged to fit sacred numerical cycles, the overarching framework of the Septuagint preserved an authentic, historically grounded memory of the Fertile Crescent's true antiquity—a memory that was severely contracted in alternative textual traditions.

The structural overlap between the Mesopotamian and biblical traditions becomes most compelling when evaluating the total duration of the antediluvian era. While the fine details and specific individual lifespans in both the Septuagint and the Sumerian King List are heavily calculated using stylized, sacred, or mathematically perfect numbers, the macro-span of the Septuagint timeline displays an uncanny alignment with real-world ancient history.

This macro-alignment is particularly remarkable when contrasted against the Sumerian King List itself. In the Mesopotamian scribal tradition, the antediluvian era spans astronomical lengths of time—typically totaling over 240,000 years across the reigns of the first eight pre-flood kings. These numbers are clearly symbolic or mathematical inflations that completely diverge from physical archaeological realities.

Rather than duplicating these vast Near Eastern timelines or severely compressing the era, the Septuagint tradition preserved a highly realistic historical interval. 

It is highly significant that the Septuagint chronology, as witnessed by Demetrius, aligns so closely with modern archaeological horizons instead of the exaggerated spans of the SKL. Because an unguided mathematical arrangement of sacred numbers would be statistically unlikely to land so close to actual physical history, this uncanny broad-stroke accuracy hints at two intriguing possibilities: it was either a case of monumental cosmic luck, or the Alexandrian redactors possessed ancient, pristine Near Eastern source records that have since been entirely lost to history.

This total of 2,264 years stands in stark contrast to the significantly compressed timelines preserved in the Masoretic Text (1,656 years) and the Samaritan Pentateuch (1,307 years). When evaluating these numbers against physical history, the Septuagint's macro-chronology reveals a striking real-world correlation.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].


==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167 / 187
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

{{RoundBoxTop}}
====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.


Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.
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'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
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=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors, as shown in the following tables.

(The '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the Black Desert of Jordan. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

{{RoundBoxTop}}
==== The Birth of Shem (A Digression) ====
Were Noah's sons born when Noah was 500 or 502?

While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.
{{RoundBoxBottom}}

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 - 1}{2} + 3^2 &amp;= 1200 + 9 \\
 &amp;= 1209
\end{aligned}
&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7^4 + 1}{2} + (7^2 - 3^2) &amp;= 1201 + 40 \\
 &amp;= 1241
\end{aligned}
&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== The Masoretic text Variation ==


[[File:Schematic_Diagram_Masoretic_Text_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Masoretic Text'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

=== Mathematical Structure of the Masoretic Chronology ===
The above diagram maps the Masoretic chronology using the identical grid framework applied to the Samaritan Pentateuch. Red circles indicate specific chronological points where the Masoretic timeline diverges from the Samaritan tradition. Key textual and numerical variations include:

==== The Universal Flood Adjustment (+349 yrs) ====
The Masoretic chronology postpones the birth of Noah and the onset of the Deluge by '''349 years''' relative to the Samaritan tradition. As outlined in the above &quot;Masoretic Adjustments&quot; section, this structural shift ensures that Noah's antediluvian ancestors do not survive past the date of the universal flood. This timeline was achieved by adding 100 years to Jared's paternal age, 120 years to Methuselah's, and 129 years to Lamech's. Consequently, these calculations place Methuselah's death precisely in the year of the Flood and Lamech's death five years prior.

==== The Smoothing of Noah's Life (-2 yrs) ====
The Masoretic chronology reduces Noah's age at the birth of Shem from 502 years (as found in the Samaritan Pentateuch) to 500 years. As discussed in the &quot;Lectio Difficilior Potior&quot; and &quot;The Birth of Shem (A Digression)&quot; sections, this change represents &quot;scribal smoothing.&quot; This editorial process rounded major milestones in Noah's life to the nearest century: his sons are born in his 500th year, the Deluge occurs in his 600th year, and his father Lamech dies in his 700th year.

==== Post-Flood Patriarchal Lifespans (-650 yrs) ====
Seven post-flood patriarchs (Arpachshad through Nahor) exhibit reduced ages at the birth of their firstborn sons within the Masoretic chronology, resulting in a cumulative deficit of 650 years compared to the Samaritan Pentateuch. Specifically, the paternal ages of Arpachshad, Shelah, Eber, Peleg, Reu, and Serug are each lower by exactly 100 years, while Nahor's age is reduced by 50 years. Given the uniform, rounded values of these variations, a direct dependency exists; either the Masoretic chronologist systematically deflated these ages, or the Samaritan chronologist systematically inflated them.

The Masoretic use of young fathering ages for these specific patriarchs may preserve an older tradition. The narrative theme of Abraham fathering Isaac at the age of 100 serves as a central literary and mathematical motif that likely predates the Babylonian exile. Consequently, the lower paternal ages in the Masoretic lineage may intentionally highlight this milestone, ensuring that a fatherhood age of 100 remains an extraordinary, miraculous exception within the broader genealogical timeline.

Under this proposed framework, Arpachshad (or Cainan) originally functioned not as a standard descendant of Shem, but as the progenitor of a distinct nine-generation patriarchal line culminating in the miraculous birth of Isaac. During the Babylonian exile, the Priestly source (&quot;P&quot;) may have adapted this older Abrahamic lineage, appending earlier primeval generations to mirror Mesopotamian creation and flood epics. Consequently, Mesopotamian archetypes—such as Adapa, Enmeduranki, and Utnapishtim—were synthesized with preexisting Hebrew traditions to form the composite figures of Adam, Enoch, and Noah. These composite primeval figures were subsequently integrated into a single, seamless genealogical chain spanning from creation to the foundations of Israel.

==== The Final Masoretic Adjustment (+215 yrs) ====
As will be explained later in this resource, it is proposed that a final 215 years was added to the Masoretic chronology to arrive at the 2666-year total. The Samaritan Pentateuch, the Septuagint, and the Book of Jubilees all claim that the Patriarchs were in Canaan for 215 years, followed by 215 years of Israelite slavery in Egypt. The Masoretic text increases the period of Egyptian slavery by 215 years, for a total of 430 years of slavery.

This increase to 430 years is difficult to reconcile with the Moses narrative, which claims that Moses was the great-grandson of Levi on the paternal line and the grandson on the maternal line. Levi entered Egypt as an adult and a free man. Moses led the children of Israel out of Egypt at eighty years old. For Moses to be a maternal grandson, his mother would have been hundreds of years old when she gave birth to him and his older siblings, Aaron and Miriam.

==== The 2800-Year to 2666-Year Transition ====
Within the Masoretic chronology, the Exodus is situated inside the third 800-year epoch, occurring precisely in the year 2666 after creation (Anno Mundi). This placement represents a deliberate scribal design choice rather than an organic chronological coincidence.

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 7^4 + 7^2 &amp;= 2401 + 49 \\
 &amp;= 2450
\end{aligned}
&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 \\
 &amp;= 2401 + 343 + 49 + 7 \\
 &amp;= 2800 \\ \text{(Base 40): } &amp; 70 \times 40 \\
 &amp;= 2800
\end{aligned}
&lt;/math&gt;

* '''Masoretic Text (Adam to Exodus):'''
:&lt;math display=&quot;block&quot;&gt;
(2800 - 46) + (349) - (2) - (650) + (215) = 2666&lt;/math&gt;
&lt;/div&gt;

As detailed in &quot;The 350-Year Symmetrical Extension,&quot; the timeline found in the Book of Jubilees likely served as the foundational model, positioning Shem at the exact midpoint of its 2,450-year chronology. The Samaritan Pentateuch appears to be a direct derivative of this Jubilees framework. Scribes systematically appended exactly 350 years to the post-Flood era, shifting the completion of the Conquest to a precise macro-structural target of 2,800 years after creation.

Envisioning the process that transitioned from the 2,800-year Samaritan chronology to the Masoretic framework presents a distinct mathematical challenge. Tracking the adjustments step-by-step from the Samaritan baseline reveals a precise transition:

* '''Samaritan Baseline (Conquest Completion):''' 2,800 years
* '''Minus Wandering and Conquest Era:''' −46 years (2,754)
* '''Plus Masoretic Flood Adjustment:''' +349 years (3,103)
* '''Minus Noah's Life Smoothing:''' −2 years (3,101)
* '''Minus Post-Flood Patriarchal Reductions:''' −650 years (2,451)
* '''The Final Masoretic Adjustment:''' +215 years (2,666)

This calculation yields exactly 2,666 years. Because individual components were dictated by rigid external theological and narrative constraints, it is surprising that the final adjustment yields exactly 2,666 years, but it also remains difficult to isolate where scribes might have executed fine-tuning to bring about this exact value and bridge the 2,800-year and 2,666-year macro-structures.

=== Mathematical Structure of the Septuagint Chronology ===
As was explained above in &quot;Septuagint Adjustments&quot;, the LXX’s editor methodically added 100 years to each pre-flood patriarch. The purpose of this adjustment was to ensure that Noah would be the exclusive survivor of the flood. As shown in the tables below, post-flood &quot;Long Chronology&quot; variations are seen to be either identical to or derived from &quot;Short Chronology&quot; values.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. The [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology shown above, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.

[[Category:Religion]]


=== Mythology ===
In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.</text>
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  </page>
  <page>
    <title>User:ThinkingScience/ND Inspired Idea Notebook</title>
    <ns>2</ns>
    <id>329177</id>
    <revision>
      <id>2845084</id>
      <parentid>2818152</parentid>
      <timestamp>2026-11-10T06:54:33Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* July 12, 2026, Sunday */ starting notes</comment>
      <origin>2845084</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9291" sha1="aw15maiy0j2qng7kfbnluwh3x1sbj9c" xml:space="preserve">Template Links:
* {{tl|Draft}}
* {{tl|underconstruction}}

'''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].'''

== &quot;Diary&quot; ==

== April 18, 2026 ==

A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a &quot;diary&quot; that will be only regarding the project.

Turned &quot;me language&quot; into expressing that everyone is welcome, that I don't &quot;own&quot; [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says &quot;I did this&quot; &quot;I did that&quot; should be gone. I think this was an improvement of some sort.

Plan for next edits on the draft page: Add an &lt;nowiki&gt;&quot; == Old Methodology needing updating == &quot;&lt;/nowiki&gt; where I will add old methodologies where I had not planned ahead too good and the &quot;Do no harm&quot; I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago.

Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future?

== April 20, 2026 ==

Why does it seem like I'm the only one using the word &quot;methodology&quot;? Did the [[Wikiversity:Research ethics]] mention it?
* I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now.

== April 21, 2026 ==

I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the &quot;Do no harm&quot;. Considering I have almost not developed anything but I still gotta work on this...to do...

== April 22, 2026 ==

I am yearning / looking forward to working on video notes in a &quot;Do no harm&quot; way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different.

== April 27, 2026 ==

I met with my father and he is a friend of the sciences. One word: hypothesis. He asked what my main hypothesis is. Of course hypothesis is a way to test if the idea is sound or if it's for the trashcan. I'm glad he gave me this feedback or interest in trying to learn more.

Today I woke up being inspired by that:

* Is there a hypothesis or a number of hypotheses related to the idea?
** Can this idea be proven false?

Why it's important to prove a hypothesis false: so that we don't waste time on trash ideas. If they are not provable we give time to the creator of the suggestion/idea to prove make a hypothesis. Only when the creator fails to provide any sort of hypothesis and maybe suggests their own idea be removed because as hard as they tried they couldn't make a functioning hypothesis...then I guess that's one of the more 'natural' ways for a project to more naturally leave Wikiversity.

Deadlines etc. can help me keep moving...

What a hypothesis is: &quot;I predict this will happen&quot; and then checking results what happens and whether it fulfilled the prediction or not and sometimes we stumble upon new things we did not expect.

[[Operationalization]] is then also needed to make &quot;ambiguous ideas&quot; measurable. My father remembers that I &quot;wanted to save the world&quot; but it was nothing other than an observation I wanted to share with other people that I had made that began years earlier. June 16 should be 2 months roughly after the creation of the Draft:Idea...

== April 29, 2026 ==

Just made my father aware of this &quot;idea&quot; that so far doesn't have a published hypothesis of any kind, not even in a &quot;basic stage&quot;. Only the deadline for June 16 exists right now.

I'm happy he did not reject me working on it. He encourages me to work harder. I got a &quot;Great job so far!&quot; compliment.

=== Google's &quot;AI Mode&quot; ===

This part was 100% AI-generated:

{{quote|What you are describing is a core concept in the philosophy of science called Falsification. It was popularized by Karl Popper, who argued that science doesn't progress by &quot;proving things right,&quot; but by rigorously trying to prove things wrong and failing to do so.}}

== April 30, 2026 ==

Was gonna start writing on the Colloquium again and composed a large message to reply/'talk to' Jtneill but I got input from &quot;AI Mode&quot; and then I realized after a while, maybe what I'm trying to ask I can find out with the help of &quot;AI Mode&quot; leading me to the right resources? ie. formatting on talk pages...how important is that? Prioritizing...

Also replying to everything? Is it really needed? Would I like to be known as &quot;Needy ThinkingScience&quot;? I can't do anything on my own? So yeah, I started thinking maybe I can focus on doing more.
Starting by talking less and the tools we have available can fix a lot anyway, LLMs are great but designed in interesting and challenging ways!

Do I even need a mentor or do I '''just think I need one'''? I am very much a needy guy in terms of hand-holding and being related to a social context &quot;historically&quot; in my life, or at least that's what I thought. What I think I need for myself may be completely wrong.

People in talk pages I interpreted as saying that I shouldn't put obstacles in my path by being mean to myself and putting pressure on myself to do certain things in a certain amount of time but I guess I couldn't stop myself! I also need a social context but would I just drag others into my &quot;bad routines&quot;? It all depends on who one interacts with...which now makes me think about &quot;the idea&quot; again. I guess I should see this as a warning sign :)

My father suggested peer-review related activities and here I am looking inward and isolating myself. I don't know...

:I figured today I begin with {{quote|April 20th experiment, &quot;AI Decisions, sure. AI-generation NEVER&quot;}}

== May 5, 2026 ==

Today I became very emotional when thinking about the importance of taking the decision to finally &quot;unload&quot; my idea on and trying to develop it on Wikiversity while talking to a friend. It's important to know when I last felt this deep of an &quot;emotion unload&quot;.

The original question from my friend in the original language: {{quote|&quot;Σκεφτόμουν εάν αυτή η συναισθηματική αντίδραση σου συμβαίνει συχνά ή εάν έτυχε απλώς τώρα στην συνάντησή μας?&quot;}} which I'll translate for you. I was also pondering the &quot;suitability&quot; of adding foreign languages to Wikiversity. It would make sense as long as that is the training material for the course which is in English about learning a foreign language.

The question as I interpret it as of this moment and which may be incorrect is this:

{{quote|I was thinking if this emotional reaction happens to you often or if it just happened now at the meeting?}}

And I think I remember my answer being that it only happened now.

It was very important for me because no longer will only I think about this draft in my own mind but other people can think too about what I describe about it. Any data I can find will also then be able to be studied by contributors or interested parties. So far, no data at all.

My point: Not having to carry this weight anymore. It was too heavy. It was basically too hard for me to formulate &quot;coherent&quot; sentences and only thought about them at brief periods while waking up or during &quot;Aha!&quot; moments.

== May 18, 2026, Monday ==

As I wrote in the diary of my main user account, screen arrived. I can now work.

I will continue work on the translation for my unfinished draft &quot;idea&quot; on the Greek Wikiversity. I'm following the &quot;Be bold&quot;. Translating also means processing, that means that a different &quot;take&quot; on the idea may happen. That may lead to a better formulated text.

== May 19, 2026, Tuesday ==

[[v:el:Χρήστης:ThinkingScience|User ThinkingScience on Greek Wikiversity]]

Today I can work on the translation and other work on Greek Wikiversity regarding my idea. This is good so I can continue the work on explaining to friends and family regarding my &quot;idea&quot;/&quot;observation&quot;. I might work right now...maybe make more notes later today?

== June 22, 2026, Monday ==

I think I'm forced to side-track due to nobody yet having shown interest in learning what my draft is about, nor interacted with me in any meaningful way yet. All initiative has come from me so it's only right it continues to come from me.
In the topic of [[Neurodiversity Movement]] I may begin to contribute.

Well I guess that I managed to contact at least 1 user. Wouldn't hurt with multiple but I guess I have to choose my &quot;battles&quot; and I only had so much &quot;mental energy&quot; for today...and it may sound silly but I need a break now...I think...

== July 05, 2026, Sunday ==

Maybe time to interact with somebody in a meaningful way? My first talk page message written by an &quot;LLM&quot;?

== July 12, 2026, Sunday ==

Google NotebookLM also exists. I should make a habit of jotting down sources. Making sure there are &quot;meta sources&quot; that control how the LLM program behaves etc. That way I can add my first Google Notebook LM reference.

== Nov 10, 2026, Tuesday ==

Google NotebookLM exists for sure but I need the 'Mind Map' functionality. It created a mind map...but probably better I use one or more of my 3 mind map apps...</text>
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    <revision>
      <id>2845087</id>
      <parentid>2845084</parentid>
      <timestamp>2026-11-10T07:26:34Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Nov 10, 2026, Tuesday */ Minder and 'standardization' in mind mapping apps</comment>
      <origin>2845087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9890" sha1="qpp685osy5izo7yq1pgitmbo962vikw" xml:space="preserve">Template Links:
* {{tl|Draft}}
* {{tl|underconstruction}}

'''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].'''

== &quot;Diary&quot; ==

== April 18, 2026 ==

A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a &quot;diary&quot; that will be only regarding the project.

Turned &quot;me language&quot; into expressing that everyone is welcome, that I don't &quot;own&quot; [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says &quot;I did this&quot; &quot;I did that&quot; should be gone. I think this was an improvement of some sort.

Plan for next edits on the draft page: Add an &lt;nowiki&gt;&quot; == Old Methodology needing updating == &quot;&lt;/nowiki&gt; where I will add old methodologies where I had not planned ahead too good and the &quot;Do no harm&quot; I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago.

Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future?

== April 20, 2026 ==

Why does it seem like I'm the only one using the word &quot;methodology&quot;? Did the [[Wikiversity:Research ethics]] mention it?
* I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now.

== April 21, 2026 ==

I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the &quot;Do no harm&quot;. Considering I have almost not developed anything but I still gotta work on this...to do...

== April 22, 2026 ==

I am yearning / looking forward to working on video notes in a &quot;Do no harm&quot; way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different.

== April 27, 2026 ==

I met with my father and he is a friend of the sciences. One word: hypothesis. He asked what my main hypothesis is. Of course hypothesis is a way to test if the idea is sound or if it's for the trashcan. I'm glad he gave me this feedback or interest in trying to learn more.

Today I woke up being inspired by that:

* Is there a hypothesis or a number of hypotheses related to the idea?
** Can this idea be proven false?

Why it's important to prove a hypothesis false: so that we don't waste time on trash ideas. If they are not provable we give time to the creator of the suggestion/idea to prove make a hypothesis. Only when the creator fails to provide any sort of hypothesis and maybe suggests their own idea be removed because as hard as they tried they couldn't make a functioning hypothesis...then I guess that's one of the more 'natural' ways for a project to more naturally leave Wikiversity.

Deadlines etc. can help me keep moving...

What a hypothesis is: &quot;I predict this will happen&quot; and then checking results what happens and whether it fulfilled the prediction or not and sometimes we stumble upon new things we did not expect.

[[Operationalization]] is then also needed to make &quot;ambiguous ideas&quot; measurable. My father remembers that I &quot;wanted to save the world&quot; but it was nothing other than an observation I wanted to share with other people that I had made that began years earlier. June 16 should be 2 months roughly after the creation of the Draft:Idea...

== April 29, 2026 ==

Just made my father aware of this &quot;idea&quot; that so far doesn't have a published hypothesis of any kind, not even in a &quot;basic stage&quot;. Only the deadline for June 16 exists right now.

I'm happy he did not reject me working on it. He encourages me to work harder. I got a &quot;Great job so far!&quot; compliment.

=== Google's &quot;AI Mode&quot; ===

This part was 100% AI-generated:

{{quote|What you are describing is a core concept in the philosophy of science called Falsification. It was popularized by Karl Popper, who argued that science doesn't progress by &quot;proving things right,&quot; but by rigorously trying to prove things wrong and failing to do so.}}

== April 30, 2026 ==

Was gonna start writing on the Colloquium again and composed a large message to reply/'talk to' Jtneill but I got input from &quot;AI Mode&quot; and then I realized after a while, maybe what I'm trying to ask I can find out with the help of &quot;AI Mode&quot; leading me to the right resources? ie. formatting on talk pages...how important is that? Prioritizing...

Also replying to everything? Is it really needed? Would I like to be known as &quot;Needy ThinkingScience&quot;? I can't do anything on my own? So yeah, I started thinking maybe I can focus on doing more.
Starting by talking less and the tools we have available can fix a lot anyway, LLMs are great but designed in interesting and challenging ways!

Do I even need a mentor or do I '''just think I need one'''? I am very much a needy guy in terms of hand-holding and being related to a social context &quot;historically&quot; in my life, or at least that's what I thought. What I think I need for myself may be completely wrong.

People in talk pages I interpreted as saying that I shouldn't put obstacles in my path by being mean to myself and putting pressure on myself to do certain things in a certain amount of time but I guess I couldn't stop myself! I also need a social context but would I just drag others into my &quot;bad routines&quot;? It all depends on who one interacts with...which now makes me think about &quot;the idea&quot; again. I guess I should see this as a warning sign :)

My father suggested peer-review related activities and here I am looking inward and isolating myself. I don't know...

:I figured today I begin with {{quote|April 20th experiment, &quot;AI Decisions, sure. AI-generation NEVER&quot;}}

== May 5, 2026 ==

Today I became very emotional when thinking about the importance of taking the decision to finally &quot;unload&quot; my idea on and trying to develop it on Wikiversity while talking to a friend. It's important to know when I last felt this deep of an &quot;emotion unload&quot;.

The original question from my friend in the original language: {{quote|&quot;Σκεφτόμουν εάν αυτή η συναισθηματική αντίδραση σου συμβαίνει συχνά ή εάν έτυχε απλώς τώρα στην συνάντησή μας?&quot;}} which I'll translate for you. I was also pondering the &quot;suitability&quot; of adding foreign languages to Wikiversity. It would make sense as long as that is the training material for the course which is in English about learning a foreign language.

The question as I interpret it as of this moment and which may be incorrect is this:

{{quote|I was thinking if this emotional reaction happens to you often or if it just happened now at the meeting?}}

And I think I remember my answer being that it only happened now.

It was very important for me because no longer will only I think about this draft in my own mind but other people can think too about what I describe about it. Any data I can find will also then be able to be studied by contributors or interested parties. So far, no data at all.

My point: Not having to carry this weight anymore. It was too heavy. It was basically too hard for me to formulate &quot;coherent&quot; sentences and only thought about them at brief periods while waking up or during &quot;Aha!&quot; moments.

== May 18, 2026, Monday ==

As I wrote in the diary of my main user account, screen arrived. I can now work.

I will continue work on the translation for my unfinished draft &quot;idea&quot; on the Greek Wikiversity. I'm following the &quot;Be bold&quot;. Translating also means processing, that means that a different &quot;take&quot; on the idea may happen. That may lead to a better formulated text.

== May 19, 2026, Tuesday ==

[[v:el:Χρήστης:ThinkingScience|User ThinkingScience on Greek Wikiversity]]

Today I can work on the translation and other work on Greek Wikiversity regarding my idea. This is good so I can continue the work on explaining to friends and family regarding my &quot;idea&quot;/&quot;observation&quot;. I might work right now...maybe make more notes later today?

== June 22, 2026, Monday ==

I think I'm forced to side-track due to nobody yet having shown interest in learning what my draft is about, nor interacted with me in any meaningful way yet. All initiative has come from me so it's only right it continues to come from me.
In the topic of [[Neurodiversity Movement]] I may begin to contribute.

Well I guess that I managed to contact at least 1 user. Wouldn't hurt with multiple but I guess I have to choose my &quot;battles&quot; and I only had so much &quot;mental energy&quot; for today...and it may sound silly but I need a break now...I think...

== July 05, 2026, Sunday ==

Maybe time to interact with somebody in a meaningful way? My first talk page message written by an &quot;LLM&quot;?

== July 12, 2026, Sunday ==

Google NotebookLM also exists. I should make a habit of jotting down sources. Making sure there are &quot;meta sources&quot; that control how the LLM program behaves etc. That way I can add my first Google Notebook LM reference.

== Nov 10, 2026, Tuesday ==

Google NotebookLM exists for sure but I need the 'Mind Map' functionality. It created a mind map...but probably better I use one or more of my 3 mind map apps...
# Minder
# VYM
# Freeplane

* What I'm looking for in this MindMap: Being able to make connections between people visually
* Having separate &quot;MindMap Islands&quot; cause I make connections between some people and not between others.
** Sometimes I have a clue that may link groups together...
* A MindMap shouldn't be this complex I need Google Notebook...

Choosing Minder for now...let's create a MindMap in this program...for &quot;MM1: Kerry Washington&quot; and &quot;MM1: Scarlett Johansson&quot;

Is MarkDown the wrong export format? It did not retain selected icon choices. Are Minder icon choices not standardized?</text>
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  </page>
  <page>
    <title>Wikiversity talk:Inactivity policy</title>
    <ns>5</ns>
    <id>330057</id>
    <revision>
      <id>2845066</id>
      <parentid>2844755</parentid>
      <timestamp>2026-11-10T03:53:55Z</timestamp>
      <contributor>
        <username>Mu301</username>
        <id>3705</id>
      </contributor>
      <minor />
      <comment>/* Votes */</comment>
      <origin>2845066</origin>
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      <format>text/x-wiki</format>
      <text bytes="13524" sha1="kt9kbbvau5gq8i8zye7afyj44ozvyds" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)
: I think this practice helps towards openness and transparency. For example, someone else in the community may become aware and comment about reasons why the SSM may be inactive. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:43, 21 July 2026 (UTC)
::I concur that the community should be made aware of, and given an opportunity to comment on, decisions that have an effect on our site. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:11, 27 August 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)
: I think 12 months is a simple, clear, appropriate timeframe for inactivity.
: I agree that one month to reply is more reasonable than 2 weeks. It is quite common for people to be on annual leave e.g., for 4 weeks. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:47, 21 July 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)
: Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC)
::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) &lt;em&gt;or&lt;/em&gt; blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:26, 16 July 2026 (UTC)
:::I agree with Justin. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:07, 19 July 2026 (UTC)
::: Yes, I agree with Koavf's suggestion; it resolves Mu301's concerns that Bureaucrat-specific actions may not be used in a 12 month period even though the user is active. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:50, 21 July 2026 (UTC)
:::: Koavf's suggestion adequately addresses my concerns. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:15, 27 August 2026 (UTC)
:: The current draft refers to administrative actions in a broad sense, not bureaucrat-specific actions. So, I think it is reasonable to interpret this as meaning that a bureaucrat would only be considered inactive if they haven't used any administrative action and haven't edited in 12 months. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:31, 21 July 2026 (UTC)
::: I concur with that interpretation. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 05:44, 7 October 2026 (UTC)

== Process ==

*For item #2 of process I would recommend requiring both talk page notification and also &quot;email this user.&quot; If someone is inactive they might not notice the talk page edit.

*For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details.
:--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:24, 16 July 2026 (UTC)
: For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC)
: In that case, go ahead. I am not against these changes. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 18 July 2026 (UTC)
:: Pinging @[[User:Mu301|Mu301]] a second time–in case you did not see my response(s). [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:03, 23 August 2026 (UTC)

==Vested interest==
Which admins could be affected by this proposal? I think that we should ping them to solicit input on a proposed policy before finalizing the terms. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 01:43, 28 August 2026 (UTC)
: User:Evolution and evolvability. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 18:46, 7 October 2026 (UTC)

== Approval ==
Is this policy good enough, or does it need additional changes? I await feedback here. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 22:52, 24 October 2026 (UTC)
: I think the proposed policy is ready (or near ready) for discussion by the Wikiversity community (e.g., by notifying via sitenotice). -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:55, 25 October 2026 (UTC)
:: Instead of the colloquium, we can hold a vote here.
:: We might also need to add a sentence per [[m:Requests for comment/Removal of advanced rights]]; if custodians are desysopped, their advanced permissions are also removed if they have any. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 21:46, 25 October 2026 (UTC)
:::I agree that it is near ready. In general, I think it is better to hold discussion on the policy talk page, rather than colloquium to make archives easier to find. I haven't (yet) reviewed the current wording. --mikeu [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:15, 28 October 2026 (UTC)
:::: I agree that it is generally better to hold policy discussions on policy talk pages because it keeps community deliberations attached to the policy. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:37, 28 October 2026 (UTC)

=== Votes ===
* {{support}} as proposer, and because after I made some final adjustments and changes, I believe it is ready to become a policy. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] /// [[User talk:Codename Noreste|⭐️✨]] 17:03, 6 November 2026 (UTC)
*{{support}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:38, 6 November 2026 (UTC)
* {{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 20:59, 6 November 2026 (UTC)
* {{support}} --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 03:53, 10 November 2026 (UTC)</text>
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  <page>
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      <text bytes="72999" sha1="teuvd9w167r0e37lyggtpmip34ivkj3" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
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This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
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{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
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gives
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
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==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
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# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
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# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
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# For additional feedback, see the following comments and [ these copy edits]
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# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main area(s) for potential improvement:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
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==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
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This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
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===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
{{{3|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
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    </revision>
    <revision>
      <id>2845034</id>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
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      <format>text/x-wiki</format>
      <text bytes="73075" sha1="7r0h1ftwy8l0xf08i5zn2p5bze4x015" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
{{{3|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
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|5=
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# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3-6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
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      <format>text/x-wiki</format>
      <text bytes="73081" sha1="pixjkkmw0vn7g32wnkcz5ywhfzjvqsv" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
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This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
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===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
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~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
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&lt;!-- Overview comments... --&gt;
# 
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
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    <revision>
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        <id>10242</id>
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      <minor />
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      <format>text/x-wiki</format>
      <text bytes="73131" sha1="bebdrinq62bf1dlkb7xrw6yyvdt3fxn" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
{{{3|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
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# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
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|5=
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|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
#* use the best psychological theory about the topic
#* provide more detailed review of the best psychological research about the topic
#* improve the quality of written expression
#* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
#* integrate more strongly with other chapters by using embedded links to related topics
#* more substantial engagement in social contribution
#* express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
#* invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
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    </revision>
    <revision>
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        <id>10242</id>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="73131" sha1="acqg1b33d02busznw832poq1yryhi3e" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
{{{3|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
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# 
|3=
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# 
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|5=
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# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
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      <format>text/x-wiki</format>
      <text bytes="73286" sha1="0zxm5ec8egttsvhkpov1golzk7bq9rj" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
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~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings) 
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings)
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
      <sha1>0zxm5ec8egttsvhkpov1golzk7bq9rj</sha1>
    </revision>
    <revision>
      <id>2845052</id>
      <parentid>2845046</parentid>
      <timestamp>2026-11-10T02:59:06Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment># Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process</comment>
      <origin>2845052</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="73592" sha1="se35ntcubyw60su68vg8vqkc1i02iuw" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
{{{3|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
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# 
|4=
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# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings) 
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings)
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
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==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
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This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
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===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# The description of problem or phenomenon is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings) 
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# The description of problem or phenomenon is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings)
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
      <sha1>tgje3iu3j219rj1my3ed90iw2x02vkq</sha1>
    </revision>
    <revision>
      <id>2845055</id>
      <parentid>2845054</parentid>
      <timestamp>2026-11-10T03:11:22Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>### To establish an abbreviation, provide the full text followed by the abbreviation in parentheses</comment>
      <origin>2845055</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="73828" sha1="0frsfehx03g070wccu0ie3pbpvrphd9" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
{{{3|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
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# 
|4=
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# 
|5=
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# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
# 
|4=
&lt;!-- Research comments... --&gt;
# 
|5=
&lt;!-- Integration comments... --&gt;
# 
|6=
&lt;!-- Conclusion comments... --&gt;
# 
|7=
&lt;!-- Style comments... --&gt;
# 
|8=
&lt;!-- Learning features comments... --&gt;
#
|9=
&lt;!-- Social contribution comments... --&gt;
#
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# The description of problem or phenomenon is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings) 
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### To establish an abbreviation, provide the full text followed by the abbreviation in parentheses
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# The description of problem or phenomenon is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings)
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### To establish an abbreviation, provide the full text followed by the abbreviation in parentheses
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
      <sha1>0frsfehx03g070wccu0ie3pbpvrphd9</sha1>
    </revision>
    <revision>
      <id>2845063</id>
      <parentid>2845055</parentid>
      <timestamp>2026-11-10T03:39:12Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>## The reference list is quite brief; deeper literature search is recommended</comment>
      <origin>2845063</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="73984" sha1="a3d5ub2tligrjv3nq4ky3i9tuu62x28" xml:space="preserve">&lt;noinclude&gt;Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit.

Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]].
__NOTOC__&lt;/noinclude&gt;&lt;includeonly&gt;
==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback==
{{RoundBoxTop|theme=8}}
This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]]
===Overall===
{{{1|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]===
{{{2|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]===
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===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]===
{{{4|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]===
{{{5|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]===
{{{6|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]===
{{{7|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]===
{{{8|No comment}}}
===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]===
{{{9|No comment}}}

{{RoundBoxBottom}}&lt;/includeonly&gt;&lt;noinclude&gt;{{collapse top|Simple example}}
==Simple example==
See also [[#Detailed example|detailed example]]
&lt;pre&gt;
&lt;!-- Official feedback --&gt;
{{MEBF/2026
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#
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#
}}
~~~~
&lt;/pre&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
#
|2=
&lt;!-- Overview comments... --&gt;
# 
|3=
&lt;!-- Theory comments... --&gt;
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|4=
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)
{{Collapse bottom}}
==Detailed example==
Example use of the template, with some common feedback comments:
&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;pre&gt;
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# The description of problem or phenomenon is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings) 
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### To establish an abbreviation, provide the full text followed by the abbreviation in parentheses
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).
## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## The reference list is quite brief; deeper literature search is recommended
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
~~~~
&lt;/pre&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;
gives
&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem
# This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem
# This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem
# This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem
# This is a basic chapter
# This is an insufficient chapter
# The main areas for potential improvement are:
## use the best psychological theory about the topic
## provide more detailed review of the best psychological research about the topic
## improve the quality of written expression
## tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]]
## integrate more strongly with other chapters by using embedded links to related topics
## more substantial engagement in social contribution
## express more in your own words—overuse of direct quotes and/or unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
## invest more time (I suspect that less than the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were invested)
&lt;!-- Overall - Edit summaries --&gt;
# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
# Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits
# [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
# The edit history is quite brief, showing limited development
# Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged
# GenAI use is acknowedged via a general statement (which has been moved to the talk page), but not by using edit summaries as required by the unit's [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
# In some places, there is overreliance on genAI
# There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]
# There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]], with a shared link to the chatbot conversation(s), when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles
# [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]].
&lt;!-- Overall – Citations --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims
# Only cite sources that you consult
# All citations need to be in the References
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
# Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes.
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [ these cop yedits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Reasonably good scenario or case study; could be more compelling
# Basic scenario or case study
# The written expression for the scenario needs improvement to be of a professional standard
# Use a more practical, real-life scenario; move review of research into a subsequent section
# Use an example that a wider audience can relate to; less than 5% of the world population are university students
# The scenario does not sufficiently illustrate the topic
# Add an engaging case study or scenario
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
# Figure 1 could be more relevant to the scenario
# Figure 1 lacks relevance to the scenario
# The Figure 1 caption should explain more clearly how it is related to the scenario
# Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section
# Figure 1 was poorly/ambiguously named; I've requested that it be renamed on Wikimedia Commons
# Include an image that illustrates the scenario/case study
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
# Put the scenario in a feature box (fixed)
&lt;!-- Overview – Explains problem --&gt;
# Clearly explains the psychological problem or phenomenon
# Explains the psychological problem or phenomenon reasonably well/in a basic way
# Briefly explains the psychological problem or phenomenon; provide more detail
# The description of problem or phenomenon is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
# Insufficient explanation of the psychological problem or phenomenon
# Clarity of written expression can be improved
# Convert the explanation of the problem from bullet-points into sentences and paragraphs
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are excellent (clear and relevant)
# The focus questions are very good/good/reasonably good/basic/promising/insufficient
# The focus questions could be improved by:
## being more specific to the topic (i.e., the sub-title)
## matching the top-level headings more closely
## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]]
## splitting double-barrelled questions into separate questions
## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## being presented in a feature box to help guide the reader (fixed)
# Add focus questions in a feature box
# See copy edits for examples of possible improvements
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Excellent—key theories are very well explained and applied
# Very good—key theories are well explained and applied; minor areas for improvements
# Good—key theories are quite well explained and applied; minor to moderate areas for improvements
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement
# A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic
# Insufficient use of relevant psychological theory about this topic
# Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
&lt;!-- Theory – Builds on --&gt;
# Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Builds on one [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
# Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Excellent/Very good use of tables, figures, and/or lists to clearly convey key theoretical information
# Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to convey key theoretical information
# The sources(s) of information presented in tables lacks sufficient citation
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
# If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source]
# If you didn't consult an original source, don't cite it
&lt;!-- Theory – Examples --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts
# Consider using more examples to illustrate theoretical concepts
# Use more examples to illustrate theoretical concepts
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research
# Excellent emphasis on systematic reviews and/or meta-analyses
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings)
# More recent research would be ideal
# Claims are well referenced
# In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
# Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags)
&lt;!-- Research – Critical thinking --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research
# Basic, formulaic
# The chapter places more emphasis on theory than on research; strive for an integrated balance
# Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence
# The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic summary and conclusion
# Insufficient as a cohesive summary of the best psychological theory and research about the topic
# Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words
# Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the  [[Motivation and emotion/Wikiversity/Edit summary|edit summary]].
# Highlights the importance of the problem or phenomenon of interest
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
# Key points are summarised in a basic way
# Summarise key points
&lt;!-- Conclusion – Focus questions --&gt;
# The focus questions are addressed
# Take-away messages for each focus question could be spelt out more clearly
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Clear take-home message(s)
# Somewhat clear take-home message(s)
# Add practical, take-home message(s)
&lt;!-- Conclusion – Word count --&gt;
# Not counted for marking purposes due to the chapter being over the maximum word count
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is excellent/very good/good/reasonably good/basic
## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]
## The quality of written expression is OK but there are several aspects which are below professional standard
## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills.
### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources
## Use active (e.g., &quot;this chapter explores&quot; or &quot;this chapter explored&quot;) rather than passive voice (e.g., &quot;this chapter will explore&quot; or &quot;this chapter has explored&quot;) [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/]
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
## Reduce excessive use of bold font
&lt;!-- Style – Sentences --&gt;
## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression
## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+.
## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence.
&lt;!-- Style – Paragraphs --&gt;
## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences.
## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences.
## Bullet points are overused. Use full sentences and paragraphs.
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st of 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]]
## &quot;Individuals&quot; is overused. &quot;People&quot; is usually a clearer term than &quot;individuals&quot;. Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally.
## Reduce use of [[w:weasel words|weasel words]]—they add bulk without improving meaning
## Use permanent, rather than relative, time references. For example, instead of &quot;20 years ago&quot;, refer to something like &quot;at the beginning of the 21st century&quot;. In this way, the text will survive better into the future, without needing to be rewritten.
## Avoid overly emotive language (e.g,. *) in science-based communication
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## The headings could be more clearly aligned with the focus questions
## The structure is overly complicated; simplify and integrate
## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion
## The chapter structure is underdeveloped; expand by using subheadings
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## Use the default heading style (e.g.,  remove additional numbers, italics, bo]ld, and/or change in font size)
## See earlier comment about [[#Heading casing|heading casing]]
## Provide more descriptive headings
## Move links from headings into their first mention in text
## Remove abbreviations/citations from headings
## Include an introductory paragraph before branching into the sub-sections
## Integrate research rather than having it as a stand-alone section(s)
## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections
## The Overview and Conclusion should not have subheadings (fixed)
## Remove colons (:) from the ends of headings
&lt;!-- Style – Grammar --&gt;
# Grammar and spelling are excellent
# Grammar
## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Improve the clarity and flow of the writing by checking and correcting use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas].
## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats')
## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who]
## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect]
## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)]
## Use past tense when describing research studies, although implications of findings could be in the present tense
## Overuse of the present perfect: use the simple past tense consistently when reporting previous research e.g.,
### Present perfect: &quot;Chivers (2010) has also suggested ...&quot;
### Simple past: &quot;Chivers (2010) suggested ...&quot;
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### Only use abbreviations such as e.g., i.e., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out (for example, that is, etcetera)
### Check and correct formatting of abbreviations (such as e.g., i.e., etc.)
### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms.
### To establish an abbreviation, provide the full text followed by the abbreviation in parentheses
### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use
### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards
### Only introduce abbreviations which are subsequently used
&lt;!-- Style – Spelling --&gt;
# Spelling
## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages.
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## &quot;Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
## Direct quotes need page numbers—even better, communicate about concepts in your own words
## Direct quotes are overused—it is better to communicate about concepts in your own words
&lt;!-- Style – Figures --&gt;
## Figures
### Very well/Well/Reasonably well captioned
### Brief captions; provide more detail to help connect the figure to the text
### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]].
### Add captions
&lt;!-- Style – Figures - Citations --&gt;
### Each Figure is referred to at least once within the main text using APA style
### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., &quot;(see Figure 1)&quot;; do not use bold, italics, check and correct capitalisation).
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Figures - Other --&gt;
### Some image uploads were removed because of a lack of sufficient/appropriate copyright information
### Numbering needs correcting
### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size)
&lt;!-- Style – Tables --&gt;
## Table(s)
### Very well/Well/Reasonably well captioned
### Brief caption(s); provide more explanatory detail about the table
### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]])
### The source(s) of the information presented in one or more tables needs acknowledgement
&lt;!-- Style – Tables - Citations --&gt;
### Each Table is referred to at least once within the main text using APA style
### Each Table is referred to at least once within the main text
### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation)
### Refer to each Table at least once within the main text (e.g., see Table 1)
&lt;!-- Style – Citations --&gt;
# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
# In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags)
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## A full stop is needed after &quot;et al&quot; (i.e., &quot;et al.&quot;) because it is an abbreviation of [[wikt:et alii|et alii]]
## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]]
## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point)
## Check and correct placement of full-stops
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources
# For citations, use APA style or wiki style, but not both
# Secondary citations are overused; strive to consult primary sources
&lt;!-- Style – References --&gt;
## References use excellent APA style
## The reference list is quite brief; deeper literature search is recommended
## References use very good/good/reasonably good/basic/poor APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-)
### Include dois where available
### Hyperlink dois for 1-click access
### Provide the full titles of journals
### Remove &quot;Retrieved from &quot;
### Use alphabetical order
### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section
### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section
### Remove bullet-points; add hanging indent
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles
# Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
# Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] &lt;!-- Internal link style --&gt;
# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]
# Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
# Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s)
# Excellent/Very good/Good/Reasonably good/Basic/No use of table(s)
# Excellent/Very good/Good/Reasonably good/Basic use of feature box(es)
# No use of feature boxes other than the opening scenario
# Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples
# No use of scenarios, case studies, or examples, other than opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
# The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end
# Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature
# The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]])
# No use of quiz(zes) and/or reflection question(s)
&lt;!-- Learning features – See also --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Also include links to related Wikipedia articles
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite
## Move external links into the [[{{PAGENAME}}#References|External links]] section
## Add more links
# [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count
&lt;!-- Learning features – External links --&gt;
# Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Use alphabetical order
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section
## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite
## Target an international audience
## Add more links
## Link to the top 3 to 6 external resources about this topic
# [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count
|9=
&lt;!-- Social contribution comments... --&gt;
# ~ logged, useful, mostly minor/minor to moderate/moderate/moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
# Thanks for the Wiki Commons uploads
# Thanks very much for your extensive contributions
# Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:47, 10 October 2026 (UTC)

==See also==
* [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]]
* [[Template:METF]]
* [[Template:MEMF]]

[[Category:Motivation and emotion/Admin/2026]]
[[Category:Motivation and emotion/Assessment/Chapter]]
&lt;/noinclude&gt;</text>
      <sha1>a3d5ub2tligrjv3nq4ky3i9tuu62x28</sha1>
    </revision>
  </page>
  <page>
    <title>Module:Sandbox/22</title>
    <ns>828</ns>
    <id>331015</id>
    <revision>
      <id>2844997</id>
      <parentid>2835243</parentid>
      <timestamp>2026-11-09T18:43:09Z</timestamp>
      <contributor>
        <username>Helpme2222</username>
        <id>3106525</id>
      </contributor>
      <origin>2844997</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="25159" sha1="qqtn6s6etgz3457r434x2kiwp824lta" xml:space="preserve">-- My function for teaching grammar

local p = {};
local frame = mw.getCurrentFrame()
local grammatical_moods = {
	&quot;Jelen ido&quot;,
	&quot;Mult ido&quot;,
	&quot;Felszolito mod&quot;,
	&quot;Felteteles mod&quot;
}
local balls = {
	grammatical_moods
}

function p.titleTest123()
	if frame.args.pagename == frame.args.ball1 or frame.args.pagename == frame.args.ball1..&quot;&amp;#039;s_Match_History&quot; then
		return &quot;Operands:&quot; .. mw.logObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;&amp;#039;s_Match_History&quot;
		}) .. &quot;...which successfully proved equal in the comparison&quot;
	else
		return &quot;Operands:&quot; .. mw.logObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;&amp;#039;s_Match_History&quot;
		}) .. &quot;...which unfortunately did not prove equal in the comparison&quot;
	end
end
		

function p.teamfightSource()
	local frame = mw.getCurrentFrame()
	
	local pageContent = mw.html.create(nil)
	
	local i = 1
	while (frame.args[&quot;team&quot;..i..&quot;_ball1&quot;]) do
		local teamTable = mw.html.create(&quot;table&quot;)
			:addClass(&quot;wikitable sortable&quot;)
			:node( mw.html.create( 'tr' )
				:node( mw.html.create( 'th' ):node(&quot;Participant&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Contribution&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Final scaling value&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Final health&quot;) )
			)
		local j = 1
		while (frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j]) do
			teamTable:node( mw.html.create( 'tr' )
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j]) ) -- ball also needs to have form and stuff.
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_score&quot;] --[[..]] --[[get contribution type]]) ) -- test: see wordpad comment
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_scaledTo&quot;] --[[..]] --[[get scaling feature]]) )
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_health&quot;]) )
				)
		end
		pageContent:node( mw.html.create(&quot;h3&quot;):node(&quot;Team &quot;..i) )
			:node( teamTable )
	end
	
	pageContent:node( mw.html.create(&quot;h2&quot;):node(&quot;Outcome&quot;) )
	local j = 1
	while (frame.args[&quot;team&quot;..frame.args.winningTeam..&quot;_ball&quot;..j]) do
		pageContent:node( --[[sprite of ball, linked to team section]] )
	end
	pageContent:node( &quot;Team&quot; )
end

local toggleButton = mw.html.create( 'span' )
	:css({
		float = &quot;right&quot;,
		[&quot;margin-right&quot;] = &quot;0&quot;,
		[&quot;font-weight&quot;] = &quot;normal&quot;
	})
	:addClass(&quot;mw-customtoggle-Infobox&quot;)
	:node(&quot;[Click to toggle]&quot;)

function p.subheading( discrim, b_content, content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node((discrim) and b_content or content)
			:node((discrim) and toggleButton)
		)
end
	
function p.s_subheading( s_content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible mw-collapsed&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node(s_content)
			:node(toggleButton)
		)
end

function p.row( discrim, name, data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; ) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(name)
		)
		:node( mw.html.create( 'td' )
			:node(data)
		)
end
	
function p.s_row( s_name, s_data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass(&quot;mw-collapsible mw-collapsed&quot;) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(s_name)
		)
		:node( mw.html.create( 'td' )
			:node(s_data)
		)
end

function p.var(symbol)
	return frame:expandTemplate{ title = &quot;Template:Var&quot;, args = {symbol} }
end

function p.iseq()
	return table.concat({
		frame.args[1] .. ( (frame.args[1] == &quot;User:Helpme2222/Sandbox/'&quot;) and &quot; is equal to User:Helpme2222/Sandbox/'&quot; or &quot; is not equal to User:Helpme2222/Sandbox/'&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;User:Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to User:Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to User:Helpme2222/Sandbox/&amp;#039;&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;Helpme2222/Sandbox/'&quot;) and &quot; is equal to Helpme2222/Sandbox/'&quot; or &quot; is not equal to Helpme2222/Sandbox/'&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to Helpme2222/Sandbox/&amp;#039;&quot;),
	}, &quot;&lt;br&gt;&quot;)
end

function p.iseqNamed()
	return table.concat({
		frame.args.arg .. ( (frame.args.arg == &quot;User:Helpme2222/Sandbox/'&quot;) and &quot; is equal to User:Helpme2222/Sandbox/'&quot; or &quot; is not equal to User:Helpme2222/Sandbox/'&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;User:Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to User:Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to User:Helpme2222/Sandbox/&amp;#039;&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;Helpme2222/Sandbox/'&quot;) and &quot; is equal to Helpme2222/Sandbox/'&quot; or &quot; is not equal to Helpme2222/Sandbox/'&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to Helpme2222/Sandbox/&amp;#039;&quot;),
	}, &quot;&lt;br&gt;&quot;)
end

function p.infoboxWBB()
	-- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html
	--[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node 
			Thankfully, these operations interpret passed nils as no-ops 
			(I'm technically using false though. Somehow that works anyways?)]]
	
	local frame = mw.getCurrentFrame()
	local ballname = frame.args[&quot;Title&quot;] or tostring(mw.title.getCurrentTitle())
	local Entities = p -- require(&quot;Module:Entities&quot;)
	
	local featureCount = tonumber(frame.args[&quot;Feature count&quot;]) or 1
	local s_featureCount = tonumber(frame.args[&quot;SUPER Feature count&quot;]) or featureCount
	local matchCount = {
		total = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local matchWinCount = {
		standard = mw.site.stats.pagesInCategory( ballname..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local s_matchCount = {
		total = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local s_matchWinCount = {
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local isSuper = 
		frame.args[&quot;SUPER Debuted&quot;] or
		frame.args[&quot;SUPER Feature count&quot;] or
		frame.args[&quot;SUPER Type&quot;] or 
		(s_matchCount.total &gt; 0) or 
		frame.args[&quot;SUPER Images&quot;] or
		nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra &quot;or nil&quot; had to be added since this is used in :addClass calls.
	-- TITLE
	local infobox = mw.html.create( 'table' )
		:addClass(&quot;wikitable&quot;)
		:css({
			float = &quot;right&quot;; 
			[&quot;margin-left&quot;] = &quot;10px&quot;
		})
		:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass((isSuper) and &quot;mw-collapsible&quot; or nil)
				:node(&quot;&lt;big&gt;&quot; .. ballname .. &quot;&lt;/big&gt;&quot;)
			)
		)
		:node((isSuper) and mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass(&quot;mw-collapsible mw-collapsed&quot;)
				:node(&quot;&lt;big&gt;&quot; .. (frame.args[&quot;SUPER Title&quot;] or (&quot;Super &quot; .. ballname)) .. &quot;&lt;/big&gt;&quot;)
			)
		)
	-- IMAGES
		:node((frame.args[&quot;Images&quot;]) and (mw.html.create( 'tr' )
			:attr(&quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot;)
			:addClass((frame.args[&quot;SUPER Images&quot;]) and &quot;mw-collapsible&quot;)
			:node( mw.html.create( 'td' )
				:attr(&quot;colspan&quot;,&quot;2&quot;)
				:node(frame.args[&quot;Images&quot;])
			))
		)
		:node((frame.args[&quot;SUPER Images&quot;]) and (mw.html.create( 'tr' )
			:attr(&quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot;)
			:addClass(&quot;mw-collapsible mw-collapsed&quot;)
			:node( mw.html.create( 'td' )
				:attr(&quot;colspan&quot;,&quot;2&quot;)
				:node(frame.args[&quot;SUPER Images&quot;])
			))
		)
	-- SECTION: Identification
		-- Section Heading
		:node( p.subheading( (frame.args[&quot;SUPER Type&quot;]), &quot;Base Identification&quot;, &quot;Basic Info&quot; ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_subheading( &quot;Super Identification&quot; ) )
		-- Type
		:node( p.row( (frame.args[&quot;SUPER Type&quot;]), &quot;Type&quot;, frame.args[&quot;Type&quot;] ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_row( &quot;Type&quot;, frame.args[&quot;SUPER Type&quot;] ) )
		-- Color
	local color = frame.args[&quot;Color&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;color&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Color&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(color ~= &quot;&quot;) and frame:expandTemplate{ title = &quot;Color&quot;, args = {color} } or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add color...&lt;/i&gt;]]&quot;
				)
			)
		)
		-- Emoji
	local emoji = frame.args[&quot;Emoji&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;emoji&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Emoji&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(emoji ~= &quot;&quot;) and ( emoji .. &quot; &lt;code&gt;&quot; .. p.emojiCodepointInternal(emoji) .. &quot;&lt;/code&gt;&quot;) or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add emoji...&lt;/i&gt;]]&quot;
				)
			)
		)
	-- SECTION: Scaling Characteristics
	--[[ Some data will be collected first. ]]
		-- (Base) Behavior
	local features = {}
	local starts = {}
	local scalings = {}
	if (featureCount &gt; 1) then
		for i = 1, featureCount do
			table.insert(features, 
				{
					frame.args[&quot;Feature &quot;..i], 
					&quot; (&quot; .. p.var(frame.args[&quot;Var &quot;..i]) .. &quot;)&quot;
				})
			table.insert(starts, 
				{
					p.var( frame.args[&quot;Var0 &quot;..i] or (frame.args[&quot;Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;, 
					frame.args[&quot;Starting value &quot;..i]
				})
			table.insert(scalings,
				{
					p.var( &quot;Δ&quot;..(frame.args[&quot;Var &quot;..i]) ) .. &quot;: &quot;,
					frame.args[&quot;Scaling &quot;..i]
				})
		end
	else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1
		features[1] = {frame.args[&quot;Feature&quot;],&quot;&quot;}
		starts[1] = {&quot;&quot;,frame.args[&quot;Starting value&quot;]}
		scalings[1] = {&quot;&quot;,frame.args[&quot;Scaling&quot;]}
	end
		-- Super Behavior
	local s_features
	local s_starts
	local s_scalings
	-- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default.
	local has_s_features;
	local has_s_starts;
	local has_s_scalings;
	if (frame.args[&quot;SUPER Feature count&quot;]) then
		if s_featureCount &gt; 1 then 
			-- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data
			s_features = mw.clone(features)
			s_starts = mw.clone(starts)
			s_scalings = mw.clone(scalings)
			-- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors.
			if s_featureCount &gt; featureCount then for i = featureCount + 1, s_featureCount do 
					table.insert( s_features, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_starts, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_scalings, {&quot;&quot;,&quot;&quot;} )
				end
			elseif s_featureCount &lt; featureCount then for i = s_featureCount + 1, featureCount do 
					s_features[i] = nil
					s_starts[i] = nil
					s_scalings[i] = nil
				end
			end
			-- Parse super data, set the has_s_... flags if triggered
			for i = 1,s_featureCount do
				if frame.args[&quot;SUPER Feature &quot;..i] then 
					s_features[i][1] = frame.args[&quot;SUPER Feature &quot;..i] 
					if (not has_s_features) then has_s_features = true end
				end
				if frame.args[&quot;SUPER Starting value &quot;..i] then 
					s_starts[i][2] = frame.args[&quot;SUPER Starting value &quot;..i] 
					if (not has_s_starts) then has_s_starts = true end
				end
				if frame.args[&quot;SUPER Scaling &quot;..i] then 
					s_scalings[i][2] = frame.args[&quot;SUPER Scaling &quot;..i] 
					if (not has_s_scalings) then has_s_scalings = true end
				end
				if frame.args[&quot;SUPER Var &quot;..i] then
					s_features[i][2] = &quot; (&quot; .. p.var(frame.args[&quot;SUPER Var &quot;..i]) .. &quot;)&quot;
					s_starts[i][1] = p.var( frame.args[&quot;SUPER Var0 &quot;..i] or (frame.args[&quot;SUPER Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;
					s_scalings[i][1] = p.var( '&lt;span style=&quot;font-style:normal&quot;&gt;Δ&lt;/span&gt;'..(frame.args[&quot;SUPER Var &quot;..i]) ) .. &quot;: &quot;
					if (not has_s_features) then has_s_features = true end
					if (not has_s_starts) then has_s_starts = true end
					if (not has_s_scalings) then has_s_scalings = true end
				end 
				s_features[i] = table.concat(s_features[i])
				s_starts[i] = table.concat(s_starts[i])
				s_scalings[i] = table.concat(s_scalings[i])
			end
			s_features = table.concat(s_features,&quot;&lt;br&gt;&quot;)
			s_starts = table.concat(s_starts,&quot;&lt;br&gt;&quot;)
			s_scalings = table.concat(s_scalings,&quot;&lt;br&gt;&quot;)
		else -- # of super features = 1? just write directly to the variable
			if frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Feature&quot;] then 
				has_s_features = true
				s_features = frame.args[&quot;SUPER Feature&quot;]
			end
			if frame.args[&quot;SUPER Var0&quot;] or frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Starting value&quot;] then 
				has_s_starts = true
				s_starts = frame.args[&quot;SUPER Starting value&quot;]
			end
			if frame.args[&quot;SUPER Scaling&quot;] then 
				has_s_scalings = true
				s_scalings = frame.args[&quot;SUPER Scaling&quot;]
			end
		end
	end
		-- finalize base data as strings now that work is otherwise complete
	if (featureCount &gt; 1) then 
		for i = 1, featureCount do
			features[i] = table.concat(features[i])
			starts[i] = table.concat(starts[i])
			scalings[i] = table.concat(scalings[i])
		end 
		features = table.concat(features,&quot;&lt;br&gt;&quot;)
		starts = table.concat(starts,&quot;&lt;br&gt;&quot;)
		scalings = table.concat(scalings,&quot;&lt;br&gt;&quot;)
	else
		features = features[1][1]
		starts = starts[1][2]
		scalings = scalings[1][2]
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (frame.args[&quot;SUPER Feature count&quot;]), &quot;Base Scaling Characteristics&quot;, &quot;Scaling Characteristics&quot; ) )
		:node( (frame.args[&quot;SUPER Feature count&quot;]) and p.s_subheading( &quot;Super Scaling Characteristics&quot; ) )
		-- Features
		:node( p.row( (has_s_features), 
				(featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			features
		) )
		:node( (has_s_features) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;SUPER Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;SUPER Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			s_features
		) )
		-- Starting values
		:node( p.row( (has_s_starts), 
				(featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;Var0&quot;] or (frame.args[&quot;Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			starts
		) )
		:node( (has_s_starts) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;SUPER Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;SUPER Var0&quot;] or (frame.args[&quot;SUPER Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			s_starts
		) )
		-- Scaling
		:node( p.row( (has_s_scalings), &quot;Scaling&quot;, scalings) )
		:node( (has_s_scalings) and p.s_row(&quot;Scaling&quot;, s_scalings) )
	-- SECTION: Career 
	--[[ Some data will be collected first. ]]
		-- (Base) Games played
	local gamesPlayedText = {}
	if matchCount.total &gt; 0 then
		table.insert(gamesPlayedText, matchCount.total .. &quot; (total)&quot;)
		if matchCount.standard &gt; 0 then table.insert(gamesPlayedText, matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- (Base) Winrate
	local winrateText = {}
	if matchCount.total &gt; 0 then
		if matchCount.standard &gt; 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
		-- Super Games played
	local s_gamesPlayedText = {}
	if s_matchCount.total &gt; 0 then
		table.insert(s_gamesPlayedText, s_matchCount.total .. &quot; (total)&quot;)
		if s_matchCount.standard &gt; 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- Super Winrate
	local s_winrateText = {}
	if s_matchCount.total &gt; 0 then
		if s_matchCount.standard &gt; 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (isSuper), &quot;Base Career&quot;, &quot;Career &quot;) )
		:node( (isSuper) and p.s_subheading( &quot;Super Career&quot;) )
		-- Debuted
		:node( p.row( (isSuper), &quot;Debut&quot;, frame.args[&quot;Debut&quot;] ) )
		:node( (isSuper) and p.s_row(&quot;Debut&quot;, frame.args[&quot;SUPER Debut&quot;]) )
		-- Games played
		:node( p.row( 
			(isSuper), 
			&quot;Games played&quot;, 
				(#gamesPlayedText &gt; 0) and table.concat(gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot; 
		) )
		:node( (isSuper) and p.s_row(
			&quot;Games played&quot;, 
				(#s_gamesPlayedText &gt; 0) and table.concat(s_gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		-- Winrate
		:node( p.row( 
			(isSuper), 
			&quot;Winrate&quot;, 
				(#winrateText &gt; 0) and table.concat(winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		:node( (isSuper) and p.s_row(
			&quot;Winrate&quot;, 
				(#s_winrateText &gt; 0) and table.concat(s_winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
	return tostring(infobox)
end

function p.referenceTest()
	local element = mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible&quot; )
		:css( &quot;text-align&quot;,&quot;left&quot; )

	local bigElement = mw.html.create( &quot;div&quot; )
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface&quot;)
		)
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface2&quot;)
		)

	return tostring(element) .. &quot;\n&quot; .. tostring(bigElement) .. &quot;\n&quot;
	
end

function p.roundedPct(num, den)
	return (num/den % 0.01 &lt; 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100)
end

function p.title()
	--[[local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;
		}
	}
	local cases2 = &quot;ba&quot;
	local cases3 = &quot;ba&quot;
	-- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = &quot;true&quot; else cases2 = &quot;false&quot; end
	-- if &quot;ANCOVA&quot; == frame:getParent():getTitle() then cases3 = &quot;true&quot; else cases3 = &quot;false&quot; end
	local detected = false
	detected = detected or true]]
	return mw.title.getCurrentTitle() 
end

function p.ifexist()
	return tonumber(frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot;, args = {frame.args[1]} })
end

function p.performance()
	for i = 1,5000000 do p.tableMake() end
	return
end

function p.performance1()
	local var = &quot;&quot;
	for i = 1,100000000 do if var[1] then end end
	return
end

function p.performance2()
	local var = &quot;&quot;
	for i = 1,100000000 do if type(var) == &quot;table&quot; then end end
	return
end

p.counter = 0
function p.tableBake()
	p.counter = p.counter + 1
	return p.counter
end


function p.tableMake()
	p.counter = 0
	local var = {
		keyword = {p.tableBake()},
		kljuczslovo = {p.tableBake()},
		yaoshiwenzi = {p.tableBake()},
		llavepalabra = {p.tableBake()},
	}
	return var
end

function p.expand()
	return frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot; }
end

function p.title3()
	return mw.title.getCurrentTitle().fullText
end

function p.getBallAttribute()
	return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4])
end

function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration )
	if _ then return mw.text.nowiki(ballData) 
	else return &quot;&quot; end
end

function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att]
	if ballData[1] then
		return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)]
	end
	return ballData
end

function p.emojiCodepointInternal( emoji )
	emoji = mw.text.trim( emoji )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.emojiCodepoint()
	local emoji = mw.text.trim( mw.getCurrentFrame().args[1] )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.title2()
	local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;,
			[&quot;Module:Sandbox/22&quot;] = &quot;Module:Sandbox/22&quot;
		}
	}
	
	local concepts = {
		grammatical_cases,
	}
	local PAGENAME = tostring(mw.title.getCurrentTitle())
	-- local detected = false
	local out = {}
	local debugger = {}
	for i, group in ipairs(concepts) do 
		if group.inessive[PAGENAME] then 
			table.insert(debugger,&quot;Okay, managed TRUE on &quot;..i..&quot;:&quot;..group.ablative)
			-- detected = detected or true
			local formattedGroup = {}
			for memberKey, memberRendervalue in pairs(group.inessive) do
				table.insert(formattedGroup, &quot;&lt;li&gt;&quot; .. memberRendervalue .. &quot;&lt;/li&gt;&quot;)
			end 
			table.insert(out, &quot;&lt;h3&gt;&quot; .. group.ablative .. &quot;&lt;/h3&gt;&lt;ul&gt;&quot; .. table.concat( formattedGroup ) .. &quot;&lt;/ul&gt;&quot; )
			table.insert(out, &quot;this sucks ass&quot;)
			return &quot;this sucks ass&quot;
		else
			table.insert(debugger,&quot;Managed FALSE on &quot;..i..&quot;:&quot;..group.ablative..&quot;. Comparator: &quot;..PAGENAME..&quot;, &quot;)
		end
	end
	return debugger
	--[[
	if detected then return tostring(detected) .. table.concat(out) end
	return
	--]]
end

function p.sanitizeChar(capture)
	-- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited
	if capture == &quot;?&quot; then
		return &quot;&quot;
	elseif capture == &quot;#&quot; then
		return &quot;No. &quot;
	end
end

function p.sanitizeTitle()
	local title = frame.args[1]
	return mw.ustring.gsub( title, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar) 
end

function p.sanitizeTitletest2()
	return mw.ustring.gsub( &quot;Ki vagy?  (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)&quot;, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar()) 
end

function p.safetyTest()
	if string.len(frame.args[1]) &lt; 500 then 
		return frame.args[1] 
	else 
		return &quot;&quot;
	end
end

--[[ 
function array_iter(t)
 	local i = 0
	return function ()
		i = i + 1
		return t[i] 
	end 
end
--]]

function p.listInline()
	--[[ This function is a godsend! 
	https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]]
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return mw.text.listToText( spritedTable , &quot;, &quot;, &quot;, and &quot; )
end

function p.listNavbox()
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return table.concat( spritedTable , &quot; • &quot;)
end

function p.listCountInternal(table) 
	local count = 0
	for index,value in ipairs( table ) do count = count + 1 end
	return count
end

function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn

function p.tableTest()
	return balls[1][1]
end

function p.todaysBall()
	--[[ https://www.lua.org/pil/3.6.html
		 The use of explicit indexing here is not strictly semantic;
		 it's just to emphasize the rotation order.
	 ]]
	local unixDay = math.floor(os.time()/86400)
	local ballTotalCount = 0
	local ballCounts = {}
    for whichTable,subtable in ipairs(balls) do 
		ballCounts[whichTable] = #subtable
		ballTotalCount = ballTotalCount + ballCounts[whichTable]
	end
	local cycleDayIndex,  cycleStartingBall = unixDay % ballTotalCount,  math.floor(unixDay/ballTotalCount) % ballTotalCount
	local cycleTodaysBall
	if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount
	elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount
	elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount
	else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end
	--[[ debug
	return cycleTodaysBall .. &quot; &quot; .. ballTotalCount .. &quot; &quot; .. os.time() .. &quot; &quot; .. math.floor(os.time()/86400) .. &quot; &quot; .. cycleDayIndex .. &quot; &quot; .. cycleStartingBall
	]]
	for whichTable = 1, #balls do 
		if ballCounts[whichTable] &gt; cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] 
		else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end
	end 
	--]]
end

return p;</text>
      <sha1>qqtn6s6etgz3457r434x2kiwp824lta</sha1>
    </revision>
    <revision>
      <id>2844999</id>
      <parentid>2844997</parentid>
      <timestamp>2026-11-09T18:51:28Z</timestamp>
      <contributor>
        <username>Helpme2222</username>
        <id>3106525</id>
      </contributor>
      <origin>2844999</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="25161" sha1="bo1qlto5a9ihcv2bdj3b7oqybb25r9t" xml:space="preserve">-- My function for teaching grammar

local p = {};
local frame = mw.getCurrentFrame()
local grammatical_moods = {
	&quot;Jelen ido&quot;,
	&quot;Mult ido&quot;,
	&quot;Felszolito mod&quot;,
	&quot;Felteteles mod&quot;
}
local balls = {
	grammatical_moods
}

function p.titleTest123()
	if frame.args.pagename == frame.args.ball1 or frame.args.pagename == frame.args.ball1..&quot;&amp;#039;s_Match_History&quot; then
		return &quot;Operands:&quot; .. mw.dumpObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;&amp;#039;s_Match_History&quot;
		}) .. &quot;...which successfully proved equal in the comparison&quot;
	else
		return &quot;Operands:&quot; .. mw.dumpObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;&amp;#039;s_Match_History&quot;
		}) .. &quot;...which unfortunately did not prove equal in the comparison&quot;
	end
end
		

function p.teamfightSource()
	local frame = mw.getCurrentFrame()
	
	local pageContent = mw.html.create(nil)
	
	local i = 1
	while (frame.args[&quot;team&quot;..i..&quot;_ball1&quot;]) do
		local teamTable = mw.html.create(&quot;table&quot;)
			:addClass(&quot;wikitable sortable&quot;)
			:node( mw.html.create( 'tr' )
				:node( mw.html.create( 'th' ):node(&quot;Participant&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Contribution&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Final scaling value&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Final health&quot;) )
			)
		local j = 1
		while (frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j]) do
			teamTable:node( mw.html.create( 'tr' )
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j]) ) -- ball also needs to have form and stuff.
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_score&quot;] --[[..]] --[[get contribution type]]) ) -- test: see wordpad comment
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_scaledTo&quot;] --[[..]] --[[get scaling feature]]) )
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_health&quot;]) )
				)
		end
		pageContent:node( mw.html.create(&quot;h3&quot;):node(&quot;Team &quot;..i) )
			:node( teamTable )
	end
	
	pageContent:node( mw.html.create(&quot;h2&quot;):node(&quot;Outcome&quot;) )
	local j = 1
	while (frame.args[&quot;team&quot;..frame.args.winningTeam..&quot;_ball&quot;..j]) do
		pageContent:node( --[[sprite of ball, linked to team section]] )
	end
	pageContent:node( &quot;Team&quot; )
end

local toggleButton = mw.html.create( 'span' )
	:css({
		float = &quot;right&quot;,
		[&quot;margin-right&quot;] = &quot;0&quot;,
		[&quot;font-weight&quot;] = &quot;normal&quot;
	})
	:addClass(&quot;mw-customtoggle-Infobox&quot;)
	:node(&quot;[Click to toggle]&quot;)

function p.subheading( discrim, b_content, content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node((discrim) and b_content or content)
			:node((discrim) and toggleButton)
		)
end
	
function p.s_subheading( s_content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible mw-collapsed&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node(s_content)
			:node(toggleButton)
		)
end

function p.row( discrim, name, data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; ) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(name)
		)
		:node( mw.html.create( 'td' )
			:node(data)
		)
end
	
function p.s_row( s_name, s_data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass(&quot;mw-collapsible mw-collapsed&quot;) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(s_name)
		)
		:node( mw.html.create( 'td' )
			:node(s_data)
		)
end

function p.var(symbol)
	return frame:expandTemplate{ title = &quot;Template:Var&quot;, args = {symbol} }
end

function p.iseq()
	return table.concat({
		frame.args[1] .. ( (frame.args[1] == &quot;User:Helpme2222/Sandbox/'&quot;) and &quot; is equal to User:Helpme2222/Sandbox/'&quot; or &quot; is not equal to User:Helpme2222/Sandbox/'&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;User:Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to User:Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to User:Helpme2222/Sandbox/&amp;#039;&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;Helpme2222/Sandbox/'&quot;) and &quot; is equal to Helpme2222/Sandbox/'&quot; or &quot; is not equal to Helpme2222/Sandbox/'&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to Helpme2222/Sandbox/&amp;#039;&quot;),
	}, &quot;&lt;br&gt;&quot;)
end

function p.iseqNamed()
	return table.concat({
		frame.args.arg .. ( (frame.args.arg == &quot;User:Helpme2222/Sandbox/'&quot;) and &quot; is equal to User:Helpme2222/Sandbox/'&quot; or &quot; is not equal to User:Helpme2222/Sandbox/'&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;User:Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to User:Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to User:Helpme2222/Sandbox/&amp;#039;&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;Helpme2222/Sandbox/'&quot;) and &quot; is equal to Helpme2222/Sandbox/'&quot; or &quot; is not equal to Helpme2222/Sandbox/'&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to Helpme2222/Sandbox/&amp;#039;&quot;),
	}, &quot;&lt;br&gt;&quot;)
end

function p.infoboxWBB()
	-- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html
	--[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node 
			Thankfully, these operations interpret passed nils as no-ops 
			(I'm technically using false though. Somehow that works anyways?)]]
	
	local frame = mw.getCurrentFrame()
	local ballname = frame.args[&quot;Title&quot;] or tostring(mw.title.getCurrentTitle())
	local Entities = p -- require(&quot;Module:Entities&quot;)
	
	local featureCount = tonumber(frame.args[&quot;Feature count&quot;]) or 1
	local s_featureCount = tonumber(frame.args[&quot;SUPER Feature count&quot;]) or featureCount
	local matchCount = {
		total = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local matchWinCount = {
		standard = mw.site.stats.pagesInCategory( ballname..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local s_matchCount = {
		total = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local s_matchWinCount = {
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local isSuper = 
		frame.args[&quot;SUPER Debuted&quot;] or
		frame.args[&quot;SUPER Feature count&quot;] or
		frame.args[&quot;SUPER Type&quot;] or 
		(s_matchCount.total &gt; 0) or 
		frame.args[&quot;SUPER Images&quot;] or
		nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra &quot;or nil&quot; had to be added since this is used in :addClass calls.
	-- TITLE
	local infobox = mw.html.create( 'table' )
		:addClass(&quot;wikitable&quot;)
		:css({
			float = &quot;right&quot;; 
			[&quot;margin-left&quot;] = &quot;10px&quot;
		})
		:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass((isSuper) and &quot;mw-collapsible&quot; or nil)
				:node(&quot;&lt;big&gt;&quot; .. ballname .. &quot;&lt;/big&gt;&quot;)
			)
		)
		:node((isSuper) and mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass(&quot;mw-collapsible mw-collapsed&quot;)
				:node(&quot;&lt;big&gt;&quot; .. (frame.args[&quot;SUPER Title&quot;] or (&quot;Super &quot; .. ballname)) .. &quot;&lt;/big&gt;&quot;)
			)
		)
	-- IMAGES
		:node((frame.args[&quot;Images&quot;]) and (mw.html.create( 'tr' )
			:attr(&quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot;)
			:addClass((frame.args[&quot;SUPER Images&quot;]) and &quot;mw-collapsible&quot;)
			:node( mw.html.create( 'td' )
				:attr(&quot;colspan&quot;,&quot;2&quot;)
				:node(frame.args[&quot;Images&quot;])
			))
		)
		:node((frame.args[&quot;SUPER Images&quot;]) and (mw.html.create( 'tr' )
			:attr(&quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot;)
			:addClass(&quot;mw-collapsible mw-collapsed&quot;)
			:node( mw.html.create( 'td' )
				:attr(&quot;colspan&quot;,&quot;2&quot;)
				:node(frame.args[&quot;SUPER Images&quot;])
			))
		)
	-- SECTION: Identification
		-- Section Heading
		:node( p.subheading( (frame.args[&quot;SUPER Type&quot;]), &quot;Base Identification&quot;, &quot;Basic Info&quot; ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_subheading( &quot;Super Identification&quot; ) )
		-- Type
		:node( p.row( (frame.args[&quot;SUPER Type&quot;]), &quot;Type&quot;, frame.args[&quot;Type&quot;] ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_row( &quot;Type&quot;, frame.args[&quot;SUPER Type&quot;] ) )
		-- Color
	local color = frame.args[&quot;Color&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;color&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Color&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(color ~= &quot;&quot;) and frame:expandTemplate{ title = &quot;Color&quot;, args = {color} } or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add color...&lt;/i&gt;]]&quot;
				)
			)
		)
		-- Emoji
	local emoji = frame.args[&quot;Emoji&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;emoji&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Emoji&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(emoji ~= &quot;&quot;) and ( emoji .. &quot; &lt;code&gt;&quot; .. p.emojiCodepointInternal(emoji) .. &quot;&lt;/code&gt;&quot;) or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add emoji...&lt;/i&gt;]]&quot;
				)
			)
		)
	-- SECTION: Scaling Characteristics
	--[[ Some data will be collected first. ]]
		-- (Base) Behavior
	local features = {}
	local starts = {}
	local scalings = {}
	if (featureCount &gt; 1) then
		for i = 1, featureCount do
			table.insert(features, 
				{
					frame.args[&quot;Feature &quot;..i], 
					&quot; (&quot; .. p.var(frame.args[&quot;Var &quot;..i]) .. &quot;)&quot;
				})
			table.insert(starts, 
				{
					p.var( frame.args[&quot;Var0 &quot;..i] or (frame.args[&quot;Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;, 
					frame.args[&quot;Starting value &quot;..i]
				})
			table.insert(scalings,
				{
					p.var( &quot;Δ&quot;..(frame.args[&quot;Var &quot;..i]) ) .. &quot;: &quot;,
					frame.args[&quot;Scaling &quot;..i]
				})
		end
	else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1
		features[1] = {frame.args[&quot;Feature&quot;],&quot;&quot;}
		starts[1] = {&quot;&quot;,frame.args[&quot;Starting value&quot;]}
		scalings[1] = {&quot;&quot;,frame.args[&quot;Scaling&quot;]}
	end
		-- Super Behavior
	local s_features
	local s_starts
	local s_scalings
	-- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default.
	local has_s_features;
	local has_s_starts;
	local has_s_scalings;
	if (frame.args[&quot;SUPER Feature count&quot;]) then
		if s_featureCount &gt; 1 then 
			-- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data
			s_features = mw.clone(features)
			s_starts = mw.clone(starts)
			s_scalings = mw.clone(scalings)
			-- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors.
			if s_featureCount &gt; featureCount then for i = featureCount + 1, s_featureCount do 
					table.insert( s_features, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_starts, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_scalings, {&quot;&quot;,&quot;&quot;} )
				end
			elseif s_featureCount &lt; featureCount then for i = s_featureCount + 1, featureCount do 
					s_features[i] = nil
					s_starts[i] = nil
					s_scalings[i] = nil
				end
			end
			-- Parse super data, set the has_s_... flags if triggered
			for i = 1,s_featureCount do
				if frame.args[&quot;SUPER Feature &quot;..i] then 
					s_features[i][1] = frame.args[&quot;SUPER Feature &quot;..i] 
					if (not has_s_features) then has_s_features = true end
				end
				if frame.args[&quot;SUPER Starting value &quot;..i] then 
					s_starts[i][2] = frame.args[&quot;SUPER Starting value &quot;..i] 
					if (not has_s_starts) then has_s_starts = true end
				end
				if frame.args[&quot;SUPER Scaling &quot;..i] then 
					s_scalings[i][2] = frame.args[&quot;SUPER Scaling &quot;..i] 
					if (not has_s_scalings) then has_s_scalings = true end
				end
				if frame.args[&quot;SUPER Var &quot;..i] then
					s_features[i][2] = &quot; (&quot; .. p.var(frame.args[&quot;SUPER Var &quot;..i]) .. &quot;)&quot;
					s_starts[i][1] = p.var( frame.args[&quot;SUPER Var0 &quot;..i] or (frame.args[&quot;SUPER Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;
					s_scalings[i][1] = p.var( '&lt;span style=&quot;font-style:normal&quot;&gt;Δ&lt;/span&gt;'..(frame.args[&quot;SUPER Var &quot;..i]) ) .. &quot;: &quot;
					if (not has_s_features) then has_s_features = true end
					if (not has_s_starts) then has_s_starts = true end
					if (not has_s_scalings) then has_s_scalings = true end
				end 
				s_features[i] = table.concat(s_features[i])
				s_starts[i] = table.concat(s_starts[i])
				s_scalings[i] = table.concat(s_scalings[i])
			end
			s_features = table.concat(s_features,&quot;&lt;br&gt;&quot;)
			s_starts = table.concat(s_starts,&quot;&lt;br&gt;&quot;)
			s_scalings = table.concat(s_scalings,&quot;&lt;br&gt;&quot;)
		else -- # of super features = 1? just write directly to the variable
			if frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Feature&quot;] then 
				has_s_features = true
				s_features = frame.args[&quot;SUPER Feature&quot;]
			end
			if frame.args[&quot;SUPER Var0&quot;] or frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Starting value&quot;] then 
				has_s_starts = true
				s_starts = frame.args[&quot;SUPER Starting value&quot;]
			end
			if frame.args[&quot;SUPER Scaling&quot;] then 
				has_s_scalings = true
				s_scalings = frame.args[&quot;SUPER Scaling&quot;]
			end
		end
	end
		-- finalize base data as strings now that work is otherwise complete
	if (featureCount &gt; 1) then 
		for i = 1, featureCount do
			features[i] = table.concat(features[i])
			starts[i] = table.concat(starts[i])
			scalings[i] = table.concat(scalings[i])
		end 
		features = table.concat(features,&quot;&lt;br&gt;&quot;)
		starts = table.concat(starts,&quot;&lt;br&gt;&quot;)
		scalings = table.concat(scalings,&quot;&lt;br&gt;&quot;)
	else
		features = features[1][1]
		starts = starts[1][2]
		scalings = scalings[1][2]
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (frame.args[&quot;SUPER Feature count&quot;]), &quot;Base Scaling Characteristics&quot;, &quot;Scaling Characteristics&quot; ) )
		:node( (frame.args[&quot;SUPER Feature count&quot;]) and p.s_subheading( &quot;Super Scaling Characteristics&quot; ) )
		-- Features
		:node( p.row( (has_s_features), 
				(featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			features
		) )
		:node( (has_s_features) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;SUPER Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;SUPER Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			s_features
		) )
		-- Starting values
		:node( p.row( (has_s_starts), 
				(featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;Var0&quot;] or (frame.args[&quot;Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			starts
		) )
		:node( (has_s_starts) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;SUPER Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;SUPER Var0&quot;] or (frame.args[&quot;SUPER Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			s_starts
		) )
		-- Scaling
		:node( p.row( (has_s_scalings), &quot;Scaling&quot;, scalings) )
		:node( (has_s_scalings) and p.s_row(&quot;Scaling&quot;, s_scalings) )
	-- SECTION: Career 
	--[[ Some data will be collected first. ]]
		-- (Base) Games played
	local gamesPlayedText = {}
	if matchCount.total &gt; 0 then
		table.insert(gamesPlayedText, matchCount.total .. &quot; (total)&quot;)
		if matchCount.standard &gt; 0 then table.insert(gamesPlayedText, matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- (Base) Winrate
	local winrateText = {}
	if matchCount.total &gt; 0 then
		if matchCount.standard &gt; 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
		-- Super Games played
	local s_gamesPlayedText = {}
	if s_matchCount.total &gt; 0 then
		table.insert(s_gamesPlayedText, s_matchCount.total .. &quot; (total)&quot;)
		if s_matchCount.standard &gt; 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- Super Winrate
	local s_winrateText = {}
	if s_matchCount.total &gt; 0 then
		if s_matchCount.standard &gt; 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (isSuper), &quot;Base Career&quot;, &quot;Career &quot;) )
		:node( (isSuper) and p.s_subheading( &quot;Super Career&quot;) )
		-- Debuted
		:node( p.row( (isSuper), &quot;Debut&quot;, frame.args[&quot;Debut&quot;] ) )
		:node( (isSuper) and p.s_row(&quot;Debut&quot;, frame.args[&quot;SUPER Debut&quot;]) )
		-- Games played
		:node( p.row( 
			(isSuper), 
			&quot;Games played&quot;, 
				(#gamesPlayedText &gt; 0) and table.concat(gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot; 
		) )
		:node( (isSuper) and p.s_row(
			&quot;Games played&quot;, 
				(#s_gamesPlayedText &gt; 0) and table.concat(s_gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		-- Winrate
		:node( p.row( 
			(isSuper), 
			&quot;Winrate&quot;, 
				(#winrateText &gt; 0) and table.concat(winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		:node( (isSuper) and p.s_row(
			&quot;Winrate&quot;, 
				(#s_winrateText &gt; 0) and table.concat(s_winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
	return tostring(infobox)
end

function p.referenceTest()
	local element = mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible&quot; )
		:css( &quot;text-align&quot;,&quot;left&quot; )

	local bigElement = mw.html.create( &quot;div&quot; )
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface&quot;)
		)
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface2&quot;)
		)

	return tostring(element) .. &quot;\n&quot; .. tostring(bigElement) .. &quot;\n&quot;
	
end

function p.roundedPct(num, den)
	return (num/den % 0.01 &lt; 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100)
end

function p.title()
	--[[local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;
		}
	}
	local cases2 = &quot;ba&quot;
	local cases3 = &quot;ba&quot;
	-- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = &quot;true&quot; else cases2 = &quot;false&quot; end
	-- if &quot;ANCOVA&quot; == frame:getParent():getTitle() then cases3 = &quot;true&quot; else cases3 = &quot;false&quot; end
	local detected = false
	detected = detected or true]]
	return mw.title.getCurrentTitle() 
end

function p.ifexist()
	return tonumber(frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot;, args = {frame.args[1]} })
end

function p.performance()
	for i = 1,5000000 do p.tableMake() end
	return
end

function p.performance1()
	local var = &quot;&quot;
	for i = 1,100000000 do if var[1] then end end
	return
end

function p.performance2()
	local var = &quot;&quot;
	for i = 1,100000000 do if type(var) == &quot;table&quot; then end end
	return
end

p.counter = 0
function p.tableBake()
	p.counter = p.counter + 1
	return p.counter
end


function p.tableMake()
	p.counter = 0
	local var = {
		keyword = {p.tableBake()},
		kljuczslovo = {p.tableBake()},
		yaoshiwenzi = {p.tableBake()},
		llavepalabra = {p.tableBake()},
	}
	return var
end

function p.expand()
	return frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot; }
end

function p.title3()
	return mw.title.getCurrentTitle().fullText
end

function p.getBallAttribute()
	return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4])
end

function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration )
	if _ then return mw.text.nowiki(ballData) 
	else return &quot;&quot; end
end

function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att]
	if ballData[1] then
		return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)]
	end
	return ballData
end

function p.emojiCodepointInternal( emoji )
	emoji = mw.text.trim( emoji )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.emojiCodepoint()
	local emoji = mw.text.trim( mw.getCurrentFrame().args[1] )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.title2()
	local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;,
			[&quot;Module:Sandbox/22&quot;] = &quot;Module:Sandbox/22&quot;
		}
	}
	
	local concepts = {
		grammatical_cases,
	}
	local PAGENAME = tostring(mw.title.getCurrentTitle())
	-- local detected = false
	local out = {}
	local debugger = {}
	for i, group in ipairs(concepts) do 
		if group.inessive[PAGENAME] then 
			table.insert(debugger,&quot;Okay, managed TRUE on &quot;..i..&quot;:&quot;..group.ablative)
			-- detected = detected or true
			local formattedGroup = {}
			for memberKey, memberRendervalue in pairs(group.inessive) do
				table.insert(formattedGroup, &quot;&lt;li&gt;&quot; .. memberRendervalue .. &quot;&lt;/li&gt;&quot;)
			end 
			table.insert(out, &quot;&lt;h3&gt;&quot; .. group.ablative .. &quot;&lt;/h3&gt;&lt;ul&gt;&quot; .. table.concat( formattedGroup ) .. &quot;&lt;/ul&gt;&quot; )
			table.insert(out, &quot;this sucks ass&quot;)
			return &quot;this sucks ass&quot;
		else
			table.insert(debugger,&quot;Managed FALSE on &quot;..i..&quot;:&quot;..group.ablative..&quot;. Comparator: &quot;..PAGENAME..&quot;, &quot;)
		end
	end
	return debugger
	--[[
	if detected then return tostring(detected) .. table.concat(out) end
	return
	--]]
end

function p.sanitizeChar(capture)
	-- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited
	if capture == &quot;?&quot; then
		return &quot;&quot;
	elseif capture == &quot;#&quot; then
		return &quot;No. &quot;
	end
end

function p.sanitizeTitle()
	local title = frame.args[1]
	return mw.ustring.gsub( title, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar) 
end

function p.sanitizeTitletest2()
	return mw.ustring.gsub( &quot;Ki vagy?  (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)&quot;, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar()) 
end

function p.safetyTest()
	if string.len(frame.args[1]) &lt; 500 then 
		return frame.args[1] 
	else 
		return &quot;&quot;
	end
end

--[[ 
function array_iter(t)
 	local i = 0
	return function ()
		i = i + 1
		return t[i] 
	end 
end
--]]

function p.listInline()
	--[[ This function is a godsend! 
	https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]]
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return mw.text.listToText( spritedTable , &quot;, &quot;, &quot;, and &quot; )
end

function p.listNavbox()
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return table.concat( spritedTable , &quot; • &quot;)
end

function p.listCountInternal(table) 
	local count = 0
	for index,value in ipairs( table ) do count = count + 1 end
	return count
end

function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn

function p.tableTest()
	return balls[1][1]
end

function p.todaysBall()
	--[[ https://www.lua.org/pil/3.6.html
		 The use of explicit indexing here is not strictly semantic;
		 it's just to emphasize the rotation order.
	 ]]
	local unixDay = math.floor(os.time()/86400)
	local ballTotalCount = 0
	local ballCounts = {}
    for whichTable,subtable in ipairs(balls) do 
		ballCounts[whichTable] = #subtable
		ballTotalCount = ballTotalCount + ballCounts[whichTable]
	end
	local cycleDayIndex,  cycleStartingBall = unixDay % ballTotalCount,  math.floor(unixDay/ballTotalCount) % ballTotalCount
	local cycleTodaysBall
	if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount
	elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount
	elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount
	else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end
	--[[ debug
	return cycleTodaysBall .. &quot; &quot; .. ballTotalCount .. &quot; &quot; .. os.time() .. &quot; &quot; .. math.floor(os.time()/86400) .. &quot; &quot; .. cycleDayIndex .. &quot; &quot; .. cycleStartingBall
	]]
	for whichTable = 1, #balls do 
		if ballCounts[whichTable] &gt; cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] 
		else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end
	end 
	--]]
end

return p;</text>
      <sha1>bo1qlto5a9ihcv2bdj3b7oqybb25r9t</sha1>
    </revision>
    <revision>
      <id>2845000</id>
      <parentid>2844999</parentid>
      <timestamp>2026-11-09T18:52:32Z</timestamp>
      <contributor>
        <username>Helpme2222</username>
        <id>3106525</id>
      </contributor>
      <origin>2845000</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="25777" sha1="slzefd24tahmlzi2npyl1yfttod0b42" xml:space="preserve">-- My function for teaching grammar

local p = {};
local frame = mw.getCurrentFrame()
local grammatical_moods = {
	&quot;Jelen ido&quot;,
	&quot;Mult ido&quot;,
	&quot;Felszolito mod&quot;,
	&quot;Felteteles mod&quot;
}
local balls = {
	grammatical_moods
}

function p.titleTest123()
	if frame.args.pagename == frame.args.ball1 or frame.args.pagename == frame.args.ball1..&quot;&amp;#039;s_Match_History&quot; then
		return &quot;Operands:&quot; .. mw.dumpObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;&amp;#039;s_Match_History&quot;
		}) .. &quot;...which successfully proved equal in the comparison&quot;
	else
		return &quot;Operands:&quot; .. mw.dumpObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;&amp;#039;s_Match_History&quot;
		}) .. &quot;...which unfortunately did not prove equal in the comparison&quot;
	end
end

function p.titleTestNoEncode()
	if frame.args.pagename == frame.args.ball1 or frame.args.pagename == frame.args.ball1..&quot;'s Match History&quot; then
		return &quot;Operands:&quot; .. mw.dumpObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;'s Match History&quot;
		}) .. &quot;...which successfully proved equal in the comparison&quot;
	else
		return &quot;Operands:&quot; .. mw.dumpObject({
			arg_pagename = frame.args.pagename, 
			arg_ball1 = frame.args.ball1, 
			concat = frame.args.ball1..&quot;'s Match History&quot;
		}) .. &quot;...which unfortunately did not prove equal in the comparison&quot;
	end
end

function p.teamfightSource()
	local frame = mw.getCurrentFrame()
	
	local pageContent = mw.html.create(nil)
	
	local i = 1
	while (frame.args[&quot;team&quot;..i..&quot;_ball1&quot;]) do
		local teamTable = mw.html.create(&quot;table&quot;)
			:addClass(&quot;wikitable sortable&quot;)
			:node( mw.html.create( 'tr' )
				:node( mw.html.create( 'th' ):node(&quot;Participant&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Contribution&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Final scaling value&quot;) )
				:node( mw.html.create( 'th' ):node(&quot;Final health&quot;) )
			)
		local j = 1
		while (frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j]) do
			teamTable:node( mw.html.create( 'tr' )
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j]) ) -- ball also needs to have form and stuff.
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_score&quot;] --[[..]] --[[get contribution type]]) ) -- test: see wordpad comment
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_scaledTo&quot;] --[[..]] --[[get scaling feature]]) )
					:node( mw.html.create( 'td' ):node(frame.args[&quot;team&quot;..i..&quot;_ball&quot;..j..&quot;_health&quot;]) )
				)
		end
		pageContent:node( mw.html.create(&quot;h3&quot;):node(&quot;Team &quot;..i) )
			:node( teamTable )
	end
	
	pageContent:node( mw.html.create(&quot;h2&quot;):node(&quot;Outcome&quot;) )
	local j = 1
	while (frame.args[&quot;team&quot;..frame.args.winningTeam..&quot;_ball&quot;..j]) do
		pageContent:node( --[[sprite of ball, linked to team section]] )
	end
	pageContent:node( &quot;Team&quot; )
end

local toggleButton = mw.html.create( 'span' )
	:css({
		float = &quot;right&quot;,
		[&quot;margin-right&quot;] = &quot;0&quot;,
		[&quot;font-weight&quot;] = &quot;normal&quot;
	})
	:addClass(&quot;mw-customtoggle-Infobox&quot;)
	:node(&quot;[Click to toggle]&quot;)

function p.subheading( discrim, b_content, content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node((discrim) and b_content or content)
			:node((discrim) and toggleButton)
		)
end
	
function p.s_subheading( s_content )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible mw-collapsed&quot; )
		:node( mw.html.create( 'th' )
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:css( &quot;text-align&quot;,&quot;left&quot; )
			:node(s_content)
			:node(toggleButton)
		)
end

function p.row( discrim, name, data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( (discrim) and &quot;mw-collapsible&quot; ) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(name)
		)
		:node( mw.html.create( 'td' )
			:node(data)
		)
end
	
function p.s_row( s_name, s_data )
	return mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass(&quot;mw-collapsible mw-collapsed&quot;) 
		:node( mw.html.create( 'td' )
			:css( &quot;font-weight&quot;,&quot;bold&quot; )
			:node(s_name)
		)
		:node( mw.html.create( 'td' )
			:node(s_data)
		)
end

function p.var(symbol)
	return frame:expandTemplate{ title = &quot;Template:Var&quot;, args = {symbol} }
end

function p.iseq()
	return table.concat({
		frame.args[1] .. ( (frame.args[1] == &quot;User:Helpme2222/Sandbox/'&quot;) and &quot; is equal to User:Helpme2222/Sandbox/'&quot; or &quot; is not equal to User:Helpme2222/Sandbox/'&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;User:Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to User:Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to User:Helpme2222/Sandbox/&amp;#039;&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;Helpme2222/Sandbox/'&quot;) and &quot; is equal to Helpme2222/Sandbox/'&quot; or &quot; is not equal to Helpme2222/Sandbox/'&quot;),
		frame.args[1] .. ( (frame.args[1] == &quot;Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to Helpme2222/Sandbox/&amp;#039;&quot;),
	}, &quot;&lt;br&gt;&quot;)
end

function p.iseqNamed()
	return table.concat({
		frame.args.arg .. ( (frame.args.arg == &quot;User:Helpme2222/Sandbox/'&quot;) and &quot; is equal to User:Helpme2222/Sandbox/'&quot; or &quot; is not equal to User:Helpme2222/Sandbox/'&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;User:Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to User:Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to User:Helpme2222/Sandbox/&amp;#039;&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;Helpme2222/Sandbox/'&quot;) and &quot; is equal to Helpme2222/Sandbox/'&quot; or &quot; is not equal to Helpme2222/Sandbox/'&quot;),
		frame.args.arg .. ( (frame.args.arg == &quot;Helpme2222/Sandbox/&amp;#039;&quot;) and &quot; is equal to Helpme2222/Sandbox/&amp;#039;&quot; or &quot; is not equal to Helpme2222/Sandbox/&amp;#039;&quot;),
	}, &quot;&lt;br&gt;&quot;)
end

function p.infoboxWBB()
	-- Passing expressions of conditional value: https://www.lua.org/pil/3.3.html
	--[[ https://www.mediawiki.org/wiki/LUAREF#mw.html:node 
			Thankfully, these operations interpret passed nils as no-ops 
			(I'm technically using false though. Somehow that works anyways?)]]
	
	local frame = mw.getCurrentFrame()
	local ballname = frame.args[&quot;Title&quot;] or tostring(mw.title.getCurrentTitle())
	local Entities = p -- require(&quot;Module:Entities&quot;)
	
	local featureCount = tonumber(frame.args[&quot;Feature count&quot;]) or 1
	local s_featureCount = tonumber(frame.args[&quot;SUPER Feature count&quot;]) or featureCount
	local matchCount = {
		total = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local matchWinCount = {
		standard = mw.site.stats.pagesInCategory( ballname..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local s_matchCount = {
		total = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History&quot;, &quot;pages&quot; ),
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard&quot;, &quot;pages&quot; ),
	}
	local s_matchWinCount = {
		standard = mw.site.stats.pagesInCategory( &quot;Super &quot; .. ballname ..&quot;'s Match History/Weapon Ball Battles/Standard/Won&quot;, &quot;pages&quot; ),
	}
	local isSuper = 
		frame.args[&quot;SUPER Debuted&quot;] or
		frame.args[&quot;SUPER Feature count&quot;] or
		frame.args[&quot;SUPER Type&quot;] or 
		(s_matchCount.total &gt; 0) or 
		frame.args[&quot;SUPER Images&quot;] or
		nil -- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. The extra &quot;or nil&quot; had to be added since this is used in :addClass calls.
	-- TITLE
	local infobox = mw.html.create( 'table' )
		:addClass(&quot;wikitable&quot;)
		:css({
			float = &quot;right&quot;; 
			[&quot;margin-left&quot;] = &quot;10px&quot;
		})
		:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass((isSuper) and &quot;mw-collapsible&quot; or nil)
				:node(&quot;&lt;big&gt;&quot; .. ballname .. &quot;&lt;/big&gt;&quot;)
			)
		)
		:node((isSuper) and mw.html.create( 'tr' )
			:node( mw.html.create( 'th' )
				:attr({
					colspan = &quot;2&quot;, 
					id = &quot;mw-customcollapsible-Infobox&quot; })
				:addClass(&quot;mw-collapsible mw-collapsed&quot;)
				:node(&quot;&lt;big&gt;&quot; .. (frame.args[&quot;SUPER Title&quot;] or (&quot;Super &quot; .. ballname)) .. &quot;&lt;/big&gt;&quot;)
			)
		)
	-- IMAGES
		:node((frame.args[&quot;Images&quot;]) and (mw.html.create( 'tr' )
			:attr(&quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot;)
			:addClass((frame.args[&quot;SUPER Images&quot;]) and &quot;mw-collapsible&quot;)
			:node( mw.html.create( 'td' )
				:attr(&quot;colspan&quot;,&quot;2&quot;)
				:node(frame.args[&quot;Images&quot;])
			))
		)
		:node((frame.args[&quot;SUPER Images&quot;]) and (mw.html.create( 'tr' )
			:attr(&quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot;)
			:addClass(&quot;mw-collapsible mw-collapsed&quot;)
			:node( mw.html.create( 'td' )
				:attr(&quot;colspan&quot;,&quot;2&quot;)
				:node(frame.args[&quot;SUPER Images&quot;])
			))
		)
	-- SECTION: Identification
		-- Section Heading
		:node( p.subheading( (frame.args[&quot;SUPER Type&quot;]), &quot;Base Identification&quot;, &quot;Basic Info&quot; ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_subheading( &quot;Super Identification&quot; ) )
		-- Type
		:node( p.row( (frame.args[&quot;SUPER Type&quot;]), &quot;Type&quot;, frame.args[&quot;Type&quot;] ) )
		:node( (frame.args[&quot;SUPER Type&quot;]) and p.s_row( &quot;Type&quot;, frame.args[&quot;SUPER Type&quot;] ) )
		-- Color
	local color = frame.args[&quot;Color&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;color&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Color&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(color ~= &quot;&quot;) and frame:expandTemplate{ title = &quot;Color&quot;, args = {color} } or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add color...&lt;/i&gt;]]&quot;
				)
			)
		)
		-- Emoji
	local emoji = frame.args[&quot;Emoji&quot;] or 
		Entities.getBallAttribute(
			&quot;Entities/Weapon Ball Battles/Balls/Data&quot;,
			ballname, 
			&quot;emoji&quot;
		)
	infobox:node( mw.html.create( 'tr' )
			:node( mw.html.create( 'td' )
				:css(&quot;font-weight&quot;, &quot;bold&quot;)
				:node(&quot;Emoji&quot;)
			)
			:node( mw.html.create( 'td' )
				:node( 
					(emoji ~= &quot;&quot;) and ( emoji .. &quot; &lt;code&gt;&quot; .. p.emojiCodepointInternal(emoji) .. &quot;&lt;/code&gt;&quot;) or
					&quot;[[Module:Entities/Weapon Ball Battles/Balls/Data|&lt;i&gt;Add emoji...&lt;/i&gt;]]&quot;
				)
			)
		)
	-- SECTION: Scaling Characteristics
	--[[ Some data will be collected first. ]]
		-- (Base) Behavior
	local features = {}
	local starts = {}
	local scalings = {}
	if (featureCount &gt; 1) then
		for i = 1, featureCount do
			table.insert(features, 
				{
					frame.args[&quot;Feature &quot;..i], 
					&quot; (&quot; .. p.var(frame.args[&quot;Var &quot;..i]) .. &quot;)&quot;
				})
			table.insert(starts, 
				{
					p.var( frame.args[&quot;Var0 &quot;..i] or (frame.args[&quot;Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;, 
					frame.args[&quot;Starting value &quot;..i]
				})
			table.insert(scalings,
				{
					p.var( &quot;Δ&quot;..(frame.args[&quot;Var &quot;..i]) ) .. &quot;: &quot;,
					frame.args[&quot;Scaling &quot;..i]
				})
		end
	else -- These weird ass tables are meant to be compatible with the clone-and-overwrite operation the supers may undergo if the SUPER feature count, unlike the hypothetical base feature count, is greater than 1
		features[1] = {frame.args[&quot;Feature&quot;],&quot;&quot;}
		starts[1] = {&quot;&quot;,frame.args[&quot;Starting value&quot;]}
		scalings[1] = {&quot;&quot;,frame.args[&quot;Scaling&quot;]}
	end
		-- Super Behavior
	local s_features
	local s_starts
	local s_scalings
	-- for some reason, html:node interprets false-valued parameters as no-op signals, while html:addClass does not. These booleans below are therefore left undefined (nil) as default.
	local has_s_features;
	local has_s_starts;
	local has_s_scalings;
	if (frame.args[&quot;SUPER Feature count&quot;]) then
		if s_featureCount &gt; 1 then 
			-- Deep copy base characteristics, i.e. use them as default, overwrite what is explicitly specified as new super data
			s_features = mw.clone(features)
			s_starts = mw.clone(starts)
			s_scalings = mw.clone(scalings)
			-- If # of super features != # of base features, trim or expand the deep-copied tables accordingly. Avoid index errors.
			if s_featureCount &gt; featureCount then for i = featureCount + 1, s_featureCount do 
					table.insert( s_features, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_starts, {&quot;&quot;,&quot;&quot;} )
					table.insert( s_scalings, {&quot;&quot;,&quot;&quot;} )
				end
			elseif s_featureCount &lt; featureCount then for i = s_featureCount + 1, featureCount do 
					s_features[i] = nil
					s_starts[i] = nil
					s_scalings[i] = nil
				end
			end
			-- Parse super data, set the has_s_... flags if triggered
			for i = 1,s_featureCount do
				if frame.args[&quot;SUPER Feature &quot;..i] then 
					s_features[i][1] = frame.args[&quot;SUPER Feature &quot;..i] 
					if (not has_s_features) then has_s_features = true end
				end
				if frame.args[&quot;SUPER Starting value &quot;..i] then 
					s_starts[i][2] = frame.args[&quot;SUPER Starting value &quot;..i] 
					if (not has_s_starts) then has_s_starts = true end
				end
				if frame.args[&quot;SUPER Scaling &quot;..i] then 
					s_scalings[i][2] = frame.args[&quot;SUPER Scaling &quot;..i] 
					if (not has_s_scalings) then has_s_scalings = true end
				end
				if frame.args[&quot;SUPER Var &quot;..i] then
					s_features[i][2] = &quot; (&quot; .. p.var(frame.args[&quot;SUPER Var &quot;..i]) .. &quot;)&quot;
					s_starts[i][1] = p.var( frame.args[&quot;SUPER Var0 &quot;..i] or (frame.args[&quot;SUPER Var &quot;..i]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) .. &quot;: &quot;
					s_scalings[i][1] = p.var( '&lt;span style=&quot;font-style:normal&quot;&gt;Δ&lt;/span&gt;'..(frame.args[&quot;SUPER Var &quot;..i]) ) .. &quot;: &quot;
					if (not has_s_features) then has_s_features = true end
					if (not has_s_starts) then has_s_starts = true end
					if (not has_s_scalings) then has_s_scalings = true end
				end 
				s_features[i] = table.concat(s_features[i])
				s_starts[i] = table.concat(s_starts[i])
				s_scalings[i] = table.concat(s_scalings[i])
			end
			s_features = table.concat(s_features,&quot;&lt;br&gt;&quot;)
			s_starts = table.concat(s_starts,&quot;&lt;br&gt;&quot;)
			s_scalings = table.concat(s_scalings,&quot;&lt;br&gt;&quot;)
		else -- # of super features = 1? just write directly to the variable
			if frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Feature&quot;] then 
				has_s_features = true
				s_features = frame.args[&quot;SUPER Feature&quot;]
			end
			if frame.args[&quot;SUPER Var0&quot;] or frame.args[&quot;SUPER Var&quot;] or frame.args[&quot;SUPER Starting value&quot;] then 
				has_s_starts = true
				s_starts = frame.args[&quot;SUPER Starting value&quot;]
			end
			if frame.args[&quot;SUPER Scaling&quot;] then 
				has_s_scalings = true
				s_scalings = frame.args[&quot;SUPER Scaling&quot;]
			end
		end
	end
		-- finalize base data as strings now that work is otherwise complete
	if (featureCount &gt; 1) then 
		for i = 1, featureCount do
			features[i] = table.concat(features[i])
			starts[i] = table.concat(starts[i])
			scalings[i] = table.concat(scalings[i])
		end 
		features = table.concat(features,&quot;&lt;br&gt;&quot;)
		starts = table.concat(starts,&quot;&lt;br&gt;&quot;)
		scalings = table.concat(scalings,&quot;&lt;br&gt;&quot;)
	else
		features = features[1][1]
		starts = starts[1][2]
		scalings = scalings[1][2]
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (frame.args[&quot;SUPER Feature count&quot;]), &quot;Base Scaling Characteristics&quot;, &quot;Scaling Characteristics&quot; ) )
		:node( (frame.args[&quot;SUPER Feature count&quot;]) and p.s_subheading( &quot;Super Scaling Characteristics&quot; ) )
		-- Features
		:node( p.row( (has_s_features), 
				(featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			features
		) )
		:node( (has_s_features) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Features&quot; or 
				( (frame.args[&quot;SUPER Var&quot;]) and (&quot;Feature (&quot; .. p.var(frame.args[&quot;SUPER Var&quot;]) .. &quot;)&quot;) or &quot;Feature&quot; ),
			s_features
		) )
		-- Starting values
		:node( p.row( (has_s_starts), 
				(featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;Var0&quot;] or (frame.args[&quot;Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			starts
		) )
		:node( (has_s_starts) and p.s_row( 
				(s_featureCount &gt; 1) and &quot;Starting values&quot; or 
				( 
					(frame.args[&quot;SUPER Var&quot;]) and (&quot;Starting value (&quot; .. ( p.var(frame.args[&quot;SUPER Var0&quot;] or (frame.args[&quot;SUPER Var&quot;]..&quot;&lt;sub&gt;0&lt;/sub&gt;&quot;) ) ) .. &quot;)&quot;) or
					&quot;Starting value&quot;
				),
			s_starts
		) )
		-- Scaling
		:node( p.row( (has_s_scalings), &quot;Scaling&quot;, scalings) )
		:node( (has_s_scalings) and p.s_row(&quot;Scaling&quot;, s_scalings) )
	-- SECTION: Career 
	--[[ Some data will be collected first. ]]
		-- (Base) Games played
	local gamesPlayedText = {}
	if matchCount.total &gt; 0 then
		table.insert(gamesPlayedText, matchCount.total .. &quot; (total)&quot;)
		if matchCount.standard &gt; 0 then table.insert(gamesPlayedText, matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- (Base) Winrate
	local winrateText = {}
	if matchCount.total &gt; 0 then
		if matchCount.standard &gt; 0 then table.insert(winrateText, p.roundedPct(matchWinCount.standard,matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
		-- Super Games played
	local s_gamesPlayedText = {}
	if s_matchCount.total &gt; 0 then
		table.insert(s_gamesPlayedText, s_matchCount.total .. &quot; (total)&quot;)
		if s_matchCount.standard &gt; 0 then table.insert(s_gamesPlayedText, s_matchCount.standard .. &quot; (standard)&quot;) end
	end
		-- Super Winrate
	local s_winrateText = {}
	if s_matchCount.total &gt; 0 then
		if s_matchCount.standard &gt; 0 then table.insert(s_winrateText, p.roundedPct(s_matchWinCount.standard,s_matchCount.standard) .. &quot;% (standard)&quot; ) end
	end
	--[[ Begin rendering ]]
		-- Section Heading
	infobox:node( p.subheading( (isSuper), &quot;Base Career&quot;, &quot;Career &quot;) )
		:node( (isSuper) and p.s_subheading( &quot;Super Career&quot;) )
		-- Debuted
		:node( p.row( (isSuper), &quot;Debut&quot;, frame.args[&quot;Debut&quot;] ) )
		:node( (isSuper) and p.s_row(&quot;Debut&quot;, frame.args[&quot;SUPER Debut&quot;]) )
		-- Games played
		:node( p.row( 
			(isSuper), 
			&quot;Games played&quot;, 
				(#gamesPlayedText &gt; 0) and table.concat(gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot; 
		) )
		:node( (isSuper) and p.s_row(
			&quot;Games played&quot;, 
				(#s_gamesPlayedText &gt; 0) and table.concat(s_gamesPlayedText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		-- Winrate
		:node( p.row( 
			(isSuper), 
			&quot;Winrate&quot;, 
				(#winrateText &gt; 0) and table.concat(winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
		:node( (isSuper) and p.s_row(
			&quot;Winrate&quot;, 
				(#s_winrateText &gt; 0) and table.concat(s_winrateText, &quot;&lt;br&gt;&quot;) or
				&quot;&lt;i&gt;Not enough data...&lt;/i&gt;&quot;
		) )
	return tostring(infobox)
end

function p.referenceTest()
	local element = mw.html.create( 'tr' )
		:attr( &quot;id&quot;,&quot;mw-customcollapsible-Infobox&quot; )
		:addClass( &quot;mw-collapsible&quot; )
		:css( &quot;text-align&quot;,&quot;left&quot; )

	local bigElement = mw.html.create( &quot;div&quot; )
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface&quot;)
		)
		:node( mw.clone(element)
			:attr( &quot;colspan&quot;, &quot;2&quot; )
			:node( &quot;shitface2&quot;)
		)

	return tostring(element) .. &quot;\n&quot; .. tostring(bigElement) .. &quot;\n&quot;
	
end

function p.roundedPct(num, den)
	return (num/den % 0.01 &lt; 0.005) and math.floor(num/den * 100) or math.ceil(num/den * 100)
end

function p.title()
	--[[local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;
		}
	}
	local cases2 = &quot;ba&quot;
	local cases3 = &quot;ba&quot;
	-- if grammatical_cases.inessive[frame:getParent():getTitle()] then cases2 = &quot;true&quot; else cases2 = &quot;false&quot; end
	-- if &quot;ANCOVA&quot; == frame:getParent():getTitle() then cases3 = &quot;true&quot; else cases3 = &quot;false&quot; end
	local detected = false
	detected = detected or true]]
	return mw.title.getCurrentTitle() 
end

function p.ifexist()
	return tonumber(frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot;, args = {frame.args[1]} })
end

function p.performance()
	for i = 1,5000000 do p.tableMake() end
	return
end

function p.performance1()
	local var = &quot;&quot;
	for i = 1,100000000 do if var[1] then end end
	return
end

function p.performance2()
	local var = &quot;&quot;
	for i = 1,100000000 do if type(var) == &quot;table&quot; then end end
	return
end

p.counter = 0
function p.tableBake()
	p.counter = p.counter + 1
	return p.counter
end


function p.tableMake()
	p.counter = 0
	local var = {
		keyword = {p.tableBake()},
		kljuczslovo = {p.tableBake()},
		yaoshiwenzi = {p.tableBake()},
		llavepalabra = {p.tableBake()},
	}
	return var
end

function p.expand()
	return frame:expandTemplate{ title = &quot;User:Helpme2222/Sandbox&quot; }
end

function p.title3()
	return mw.title.getCurrentTitle().fullText
end

function p.getBallAttribute()
	return p.getBallAttributeInternal(frame.args[1], frame.args[2], frame.args[3], frame.args[4])
end

function p.getBallAttributeInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local _, ballData = pcall( p.getBallAttributeInternalInternal, arg1_table, arg2_ball, arg3_att, arg4_iteration )
	if _ then return mw.text.nowiki(ballData) 
	else return &quot;&quot; end
end

function p.getBallAttributeInternalInternal(arg1_table, arg2_ball, arg3_att, arg4_iteration)
	local ballData = mw.loadData(arg1_table)[arg2_ball][arg3_att]
	if ballData[1] then
		return ballData[tonumber(arg4_iteration) or p.listCountInternal(ballData)]
	end
	return ballData
end

function p.emojiCodepointInternal( emoji )
	emoji = mw.text.trim( emoji )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.emojiCodepoint()
	local emoji = mw.text.trim( mw.getCurrentFrame().args[1] )
	local i,v = mw.ustring.codepoint( emoji, 1, mw.ustring.len(emoji)  )
	if v then return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) .. &quot; U+&quot; .. string.format(&quot;%X&quot;, v )
	else return &quot;U+&quot; .. string.format(&quot;%X&quot;, i ) end
end

function p.title2()
	local grammatical_cases = {
		ablative = 1,
		inessive = {
			ANCOVA = &quot;ANCOVA&quot;,
			[&quot;Module:Sandbox/22&quot;] = &quot;Module:Sandbox/22&quot;
		}
	}
	
	local concepts = {
		grammatical_cases,
	}
	local PAGENAME = tostring(mw.title.getCurrentTitle())
	-- local detected = false
	local out = {}
	local debugger = {}
	for i, group in ipairs(concepts) do 
		if group.inessive[PAGENAME] then 
			table.insert(debugger,&quot;Okay, managed TRUE on &quot;..i..&quot;:&quot;..group.ablative)
			-- detected = detected or true
			local formattedGroup = {}
			for memberKey, memberRendervalue in pairs(group.inessive) do
				table.insert(formattedGroup, &quot;&lt;li&gt;&quot; .. memberRendervalue .. &quot;&lt;/li&gt;&quot;)
			end 
			table.insert(out, &quot;&lt;h3&gt;&quot; .. group.ablative .. &quot;&lt;/h3&gt;&lt;ul&gt;&quot; .. table.concat( formattedGroup ) .. &quot;&lt;/ul&gt;&quot; )
			table.insert(out, &quot;this sucks ass&quot;)
			return &quot;this sucks ass&quot;
		else
			table.insert(debugger,&quot;Managed FALSE on &quot;..i..&quot;:&quot;..group.ablative..&quot;. Comparator: &quot;..PAGENAME..&quot;, &quot;)
		end
	end
	return debugger
	--[[
	if detected then return tostring(detected) .. table.concat(out) end
	return
	--]]
end

function p.sanitizeChar(capture)
	-- If the set of likely illegal characters to appear in title expands, a rewrite of this as a table is merited
	if capture == &quot;?&quot; then
		return &quot;&quot;
	elseif capture == &quot;#&quot; then
		return &quot;No. &quot;
	end
end

function p.sanitizeTitle()
	local title = frame.args[1]
	return mw.ustring.gsub( title, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar) 
end

function p.sanitizeTitletest2()
	return mw.ustring.gsub( &quot;Ki vagy?  (🌡️🌾⭐🗡️ VS 🪐 🪨 💣 🧿)&quot;, &quot;(?:\\?|\\#)&quot;, p.sanitizeChar()) 
end

function p.safetyTest()
	if string.len(frame.args[1]) &lt; 500 then 
		return frame.args[1] 
	else 
		return &quot;&quot;
	end
end

--[[ 
function array_iter(t)
 	local i = 0
	return function ()
		i = i + 1
		return t[i] 
	end 
end
--]]

function p.listInline()
	--[[ This function is a godsend! 
	https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#mw.text.listToText --]]
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return mw.text.listToText( spritedTable , &quot;, &quot;, &quot;, and &quot; )
end

function p.listNavbox()
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.concat
	local spritedTable = {}
    for i,entry in ipairs(balls[tonumber(frame.args[1])]) do
	    spritedTable[i] = &quot;{{Sprite|named=1|linked=1|&quot;..entry..&quot;}}&quot;
    end
	return table.concat( spritedTable , &quot; • &quot;)
end

function p.listCountInternal(table) 
	local count = 0
	for index,value in ipairs( table ) do count = count + 1 end
	return count
end

function p.listCount() return table.maxn(balls[tonumber(frame.args[1])]) end
	-- https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#table.maxn

function p.tableTest()
	return balls[1][1]
end

function p.todaysBall()
	--[[ https://www.lua.org/pil/3.6.html
		 The use of explicit indexing here is not strictly semantic;
		 it's just to emphasize the rotation order.
	 ]]
	local unixDay = math.floor(os.time()/86400)
	local ballTotalCount = 0
	local ballCounts = {}
    for whichTable,subtable in ipairs(balls) do 
		ballCounts[whichTable] = #subtable
		ballTotalCount = ballTotalCount + ballCounts[whichTable]
	end
	local cycleDayIndex,  cycleStartingBall = unixDay % ballTotalCount,  math.floor(unixDay/ballTotalCount) % ballTotalCount
	local cycleTodaysBall
	if ballTotalCount % 7 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*7) % ballTotalCount
	elseif ballTotalCount % 11 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*11) % ballTotalCount
	elseif ballTotalCount % 13 ~= 0 then cycleTodaysBall = (cycleStartingBall + cycleDayIndex*13) % ballTotalCount
	else cycleTodaysBall = (cycleStartingBall + cycleDayIndex*(ballTotalCount - 1)) % ballTotalCount end
	--[[ debug
	return cycleTodaysBall .. &quot; &quot; .. ballTotalCount .. &quot; &quot; .. os.time() .. &quot; &quot; .. math.floor(os.time()/86400) .. &quot; &quot; .. cycleDayIndex .. &quot; &quot; .. cycleStartingBall
	]]
	for whichTable = 1, #balls do 
		if ballCounts[whichTable] &gt; cycleTodaysBall then return balls[whichTable][cycleTodaysBall + 1] 
		else cycleTodaysBall = cycleTodaysBall - ballCounts[whichTable] end
	end 
	--]]
end

return p;</text>
      <sha1>slzefd24tahmlzi2npyl1yfttod0b42</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2026/Moral disgust and jury decision-making</title>
    <ns>0</ns>
    <id>331110</id>
    <revision>
      <id>2845047</id>
      <parentid>2840628</parentid>
      <timestamp>2026-11-10T00:38:29Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Copyediting</comment>
      <origin>2845047</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28916" sha1="7i522gdd0nuwn3s644gx1mviad5py02" xml:space="preserve">{{title|Moral disgust and jury decision-making:&lt;b&gt;How does moral disgust impact jurors' judgements of guilt, blame, and punishment?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Trial by Jury - Chaos in the Courtroom.png|thumb|right|250px |'''Figure 1'''. A jury is a group of sworn individuals who determine a defendant's guilt. ]]

Consider sitting on a jury. The defendant appears composed, well-dressed, and speaks clearly. Someone like this couldn't have done anything bad — that's your instinct.

Then you hear the charge: embezzlement. Over three years, they quietly transferred company funds to a personal account. Thousands of dollars were spent, and real people were harmed. You are angry. It's wrong. However, it does not make you feel sick.

Consider the same composed defendant in the same courtroom, but this time the charge is posing as a charity volunteer to gain the trust of dying hospice patients before stealing the donations intended for their care. The financial loss is the same as the embezzlement case. Same amount, same number of victims.

But something else arises in you. Not just anger, but more of a revulsion, as if something was contaminated. This doesn't simply feel like a crime. It has an unclean feeling to it.

The same defendant. Same harm. So, why does one case make you feel cold and angry, while the other makes your skin crawl? And, aside from the actual harm done, could that visceral reaction be influencing how harshly you believe they should be punished?
{{RoundBoxBottom}}

Across the world, juries have varying degrees of responsibility to establish a verdict of a case based on evidence presented (Hans, 2008). However, moral disgust, an emotional gut reaction, seems to influence how jurors assess guilt, blame and punishment, regardless of the strength of the evidence presented (Salerno and Peter-Hagene, 2013). A disgusted juror may unconsciously become more certain of guilt, assign more blame and suggest harsher punishment, relying less on what the facts of the case provide. This raises questions for current legal practices. Courts commonly admit gruesome evidence under the assumption that its evidentiary value outweighs its emotional impact, an assumption this {{which}} body of research targets (Bright and Goodman-Delahunty, 2006). Psychological science can provide the legal system with theoretical evidence about the nature and causes of this bias.

{{Display box|title=Focus questions|titleBgColor=#006499|contentBgColor=#f8f9fa|linkColor=blue|content=
* What is moral disgust, and how can it be separated from physical disgust or anger?
* How do intuitive reactions and slow deliberation interact when jurors determine guilt?
* How may moral disgust impact blame?
* How might moral disgust impact how punitive jurors are?}}

== What is moral disgust? ==
When individuals witness moral transgression, a feeling of revulsion is experienced. It shares a resemblance to the physiological and psychological mechanisms of core disgust {{ic|explain what is meant by &quot;core disgust&quot;}}. Researchers are continuing to understand the evolutionary and cognitive overlap between these interrelated emotions (Chapman &amp; Anderson, 2012). Jurors are often tasked with the role of judging morally repugnant acts, which means that understanding how the mechanisms of moral disgust work in comparison to the current research on core disgust within the legal system is vital. 

=== Disgust ===
[[w:Disgust|Disgust]] is one of the six universal emotions that humans experience and can be visually recognised on a person's face (Ekman, 1992). The emotion of disgust is theorised to have originated from a biological protective mechanism to protect an organism from harm that may be brought on by food or poison (Rozin &amp; Fallon, 1987). Disgust has evolved to become an emotion/behaviour directed at anything that may potentially cause harm or disease. For example, 

* body products 
* animals 
* hygiene 
* body violations, death or disease 

Paul Ekman {{f}} proposed six basic emotions in his literature based on the theories that these emotions, across research all over the world, {{g}} concluded in agreement on six emotions: happiness, surprise, fear, anger, sadness and disgust. Therefore, he proposed that the universality of these findings and the facial expression alone could conclude that there are 6 basic emotions (Ekman, 1992).

=== Moral disgust ===
Moral disgust refers to the emotional response a person feels when their social or cultural norms are transgressed, therefore resulting in a reaction {{f}}. Moral disgust is complex, and there is varying evidence for and against its genuine purpose as an emotion. Researchers argue that moral disgust is a metaphorical extension of physical disgust; shared facial expression, language, etc. Other researchers argue that it's functionally unique to traditional, physical disgust (Chapman &amp; Anderson, 2012). It does not reliably trigger the same physiological processes: facial muscle activation, nausea, etc.  

Moral disgust is triggered by acts that are supposedly a violation of an individual's perceived purity or dignity {{f}}. For example, betraying someone's trust (cheating on a partner) or exploiting a vulnerable person (as mentioned in the scenario at the start) {{g}}. 

Rozin &amp; Fallon (1999) proposed a theory that moral disgust specifically clusters around violations of a person's divinity or purity. This is different from anger, which generally clusters around autonomy violations, {{g}} also opposed to contempt, which clusters around community violations. This distinction is vital when discussing emotion for jurors. If moral disgust behaves differently from physical disgust, it may influence judgement in the courtroom through a different pathway than a simple reaction to physically disgusting evidence.

Russell &amp; Giner-Sorolla (2013) discuss four competing arguments of {{g}} disgust's role in moral condemnation. Firstly, the most flexible view of moral disgust that can be applied widely for many crimes is the general morality position. It claims that disgust is an all-purpose emotion which is seen through different kinds of moral disapproval, not just a narrow category {{example}}. Secondly, disgust as a violation of bodily and spiritual purity is more specific. As mentioned prior, this claim is supported by Rozin's CAD (contempt, anger, disgust) triad (Rozin et al., 1999) {{example}}. The third argument proposes that moral disgust is only a metaphor. The true emotion being experienced is anger, and individuals just use language as a metaphor (e.g., &quot;that's disgusting&quot;) for how condemning they feel {{ex}}. Finally, Russell &amp; Giner-Sorolla's own claim for moral disgust is a middle ground for all the arguments in their research. Regardless of whether harm or justice is involved, they contend that disgust specifically tracks violations pertaining to the body and its products {{ex}}.

A study involving 148 {{who}} participants {{where}} ranking their emotions on a scale (Ouellet-Courtois &amp; Radomsky, 2023) found that moral disgust elicited as much as physical disgust did {{g}}. This supports a claim that moral disgust is not a watered-down, metaphorical use of an emotion but a genuine, physical feeling of disgust. It was also revealed that moral disgust uniquely triggered a significantly higher feeling of mental contamination (feeling immoral) than core disgust did. This internalised feeling about your own moral state just from witnessing an immoral act, ties back to Russell &amp; Giner-Sorolla's bodily-violation pattern and strengthens Rozin &amp; Fallon's (1999) definition of moral disgust {{g}}.

&lt;quiz display=&quot;simple&quot;&gt;
{According to Russell and Giner-Sorolla (2013), which of the following is NOT one of the arguments presented regarding the role of disgust in moral condemnation? :
|type=&quot;()&quot;}
- General Morality Position
- Bodily and Spirituality Position
- Metaphorical Position
+ Universal Justice Position
&lt;/quiz&gt;

== Moral disgust and juror judgements ==
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A jury is a group of individuals that {{g}} are selected to make a decision based on evidence. They are picked to be impartial and unbiased opinions that aid in the determining of a person's guilt, how blame-worthy they are and the punishment they deserve {{g}}. Jurors are asked to make judgements based on the evidence presented to them in court. However, emotions play a large, unsuspecting role in determining the guilt of a person. Moral disgust in particular seems to have a large impact on the way that jurors perceive and shape jurors' judgements.
{{RoundBoxBottom}}

=== Juror judgements ===
A juror's role internationally is consistently to be {{aw}} an impartial decision-maker based solely on presented evidence. Despite the country or jurisdiction, jurors are required to surpass psychological challenges of letting emotion get in the way of this job {{f}}. However, research commonly suggests that jurors often let their emotions influence or cloud their judgement (Feigenson, 2016). This section looks at how disgust influences three agencies in decision-making as a juror. Guilt, blame and punishment {{g}}. Moral disgust is a complex social emotion which forces researches to measure using standard, operationalised measures for disgust {{g}}. It is undeniably present in decision-making for jurors, but it is often collapsed into a general category of disgust {{f}}. Moral disgust is underrepresented in terminology and research. 

=== Guilt ===
[[File:Masaccio - The Expulsion from the Garden of Eden (detail) - WGA14180.jpg|right|thumb|290x290px|Figure 3: Expulsion from the Garden of Eden; Adam and Eve experiencing the guilt of {{g}} their actions.]]

The [[w:Dual_process_theory_(moral_psychology)|dual-process theory]] argues that thinking can be separated into two distinct types (Evans &amp; Stanovich, 2013). A {{g}} Type 1 system involves autonomous processing that is often responsible for producing intuitive responses without the need for working memory. Type 2 is often more reflective and involves more explicit cognitive processes alongside working memory. A key feature of Type 2 processing is cognitive decoupling and mental simulation. Cognitive decoupling is an ability to separate what you are assuming from what you believe to be true, and mental simulation involves creating hypothetical scenarios and outcomes within your mind (Evans &amp; Stanovich, 2013). Supporting evidence suggests that disgust is a Type 1 output, automatic and arriving before deliberate reasoning can take place {{f}}.  

Disgust influences juror processing through distinct mechanisms. Altered evidentiary standards are the impact of disgust lowering the threshold of proof required for conviction. Thus leading to a biassed information search {{g}}. Jurors will selectively focus on specific details that support their emotion, becoming more certain of their decision {{f}}. Other emotions, for example, anxiety, are generally associated with more uncertainty, which leads to more systematic information processing (Feigenson, 2016). Saddened by emotional testimony, mock jurors in a reckless-driving criminal case were able to identify inconsistencies more precisely {{ic|than?}}, which is a sign of systematic information processing (Semmler &amp; Brewer, 2002).   

Additionally, Bright &amp; Goodman-Delahunty's (2006) research discusses a person's use of their inner emotional states as a way to reason with information they are being given {{expand}}. This aligns with the affect {{sp}} that is being theorised {{ic|by who?}}. Emotional pathways generally interact simultaneously, and their distinct mechanisms will be vital in understanding the relationship between cognition and judgementn {{vague}}. Feigenson (2016) also notes that jurors may have the ability to override their initial reactions through discussion and reasoning {{expand}}.

=== Blame ===
According to Heider's (1958) [[w:Attribution_(psychology)|attribution theory]], individuals naturally evaluate behaviour by balancing personal casuality against situational causality to determine intentionality. In a legal setting, jurors have the expectation that they will objectively weigh these factors fairly and decide a defendant's blame-worthiness. However, cases that evoke intense disgust can interrupt the rational assessment that jurors are expected to undertake. Instead of evaluating control before assigning blame, the process that Heider's (1958) theory outlines is that jurors often invert this sequence through a phenomenon introduced by Alicke (2000). 

The core idea for the Culpable Control Model (CCM) is an explanation of how people make judgements about blame and responsibility. Alicke (2000) argues that judgement is not always subjective, as blame does not solely rely on evidence about a person's actions and control over the action but also on individuals' emotions, values and assumptions. The assumption of the CCM is that people form a blame judgement first, based on actions, and then interpret the evidence on how it is supported second (Alicke, 2000). The main mechanism of this theory is that evidence isn't evaluated and blame is placed. Emotional reactions, values, assumptions and expectations of people as humans can heavily shape how evidence is interpreted in the first place (Alicke, 2000). According to CCM, processing for a juror operates firstly with spontaneous evaluation, then blame motive then retroactive rationalisation {{g}}. If people search for evidence to support an assumption based on their emotions, based on Alicke's (2000) claim, disgust would act in the same way.

&lt;quiz display=&quot;simple&quot;&gt;
{According to Mark Alicke’s Culpable Control Model (CCM), what is the chronological sequence of a juror's cognitive processing when assigning blame?:
|type=&quot;()&quot;}
- Evidence evaluation --&gt; control assessment --&gt;spontaneous blame judgment.
+ Spontaneous evaluation --&gt; blame motive --&gt;retroactive rationalisation.
&lt;/quiz&gt;

=== Punishment ===
Classic social identity theory posits that groups and whole organisations that possess the same cultural and social norms will perceive themselves as sharing something in common (Tajfel &amp; Turner, 1979). This adds to their own personal self-concept as an individual and as a member of a society/societies. Bornewasser &amp; Bober (1987) critique that this theory describes a class or collective of people (people who may share traits but have no real interdependence) but not a genuine societal group. Groups depend on structure, roles, positions and relationships that require dependence from people.

[[File:Baxter,Social Identity Theory- Nov 2016.png|thumb|390x390px|Figure 2: Flowchart of social identity theory]]

Through this theory, a jury is considered to be a genuine structured group. There are roles, a task and an interdependence through deliberation. This theory may offer a further point on why moral disgust can bias jurors. Moral disgust may be a function of a social categorisation cue, marking defendants as the out-group rather than in the jury's moral in-group. Bornewasser &amp; Bober (1987), {{g}} claim that even arbitary group labels are enough to produce an in-group favouritism and out-group deterrence. If a defendant's act triggers moral disgust, this may mark them as being in the moral &quot;outgroup&quot; in a juror's mind. Independent of the actual evidence being presented, this compounds moral disgust biases. 

Additionally, Feigenson (2016) discusses the role of integral and incidental emotions on the consequent punishment of a defendant. There are various external factors that will cause a juror to be more punitive due to the emotions they are experiencing. Regardless of the discouragement for using emotional decision making, jurors cannot always keep their habits of thinking and feeling in check (Feigenson, 2016). This can be reworked into Evans and Stanovich's (2013) research about processing systems involved in emotions and thinking. Type 1 processes of thinking is {{g}} assumed to be heavily reliant on predetermined values and morals which a juror will rarely think over before making a decision. It is thus assumed that Type 2 processing allows for a more deliberate consideration system when accounting for consequences and outcomes (Evans and Stanovich, 2013). This does not, however, guarantee better judgement to use when making heavily implicating decisions. From a cognitive perspective, disgust is not an irrational feeling. The mind uses emotions to understand and prcoess information (Salerno, 2021). The brain will often treat the intensity of an emotion as a barometer to measure their belief of the harm that has been done. This mechanism has been mentioned previously, so the theory is supporting the claim that the more revulsion a person feels, the more justification they have for more severe punishment (Salerno, 2021; Feigenson, 2016).  

== Additional evidence ==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Empirical evidence ===
Early research intially found a powerful link between physical taste and moral judgement. The original study that was conducted was with a small sample of 57 participants {{f}}. It found that consuming a bitter beverage before reading scenarios caused signficantly harsher judgement on the morality of the scenarios (Eskine, 2011). Research by Ghelfi et al, {{g}} (2020) suggests that physical and moral disgust may have overlapping processes but the research failed to replicate Eskine's (2011) study. It was replicated across 11 different laboratories with 1,137 participants to test whether bitter beverages led to harsher moral judgements than sweet beverages or water. Ghelfi's (2020) research discusses to {{g}} impacts of different sensory systems on the impact of moral judgements. However the replication study produced neglible effects.  

=== Emotional response patterns ===
Fishbein &amp; Ajzen's (1975) [[w:Expectancy-value_theory|expectancy-value model]] states that a person's given attitude towards an object is a direct function of the value that a person attaches to that object's attributes or outcomes. In explicit terms, a person's value on an object will be a direct function of the value that they place on said {{aw}} item and the positive or negative emotions they associate with it. Furthermore, the positive or negative characteristics that they associate with the object will be a weighted decision on the direct outcome that may occur due to the object.

In a review of the role of emotion-induced expectancies, Desteno et al. (2004) argues that positive and negative emotions have been grouped together and oversimplified, which makes distinguishing them difficult. The study found that specificity operates through an emotion-matching process. Whilst the study did not touch on disgust specifically, it found that persuasion about a situation increased when a person's current emotional state matched the emotional framing of a message (Desteno et al. (2004). The broader emotion-specificity principle suggests a comparable mechanism that may apply to disgust. {{ex}}

=== Integral versus incidental emotion distinction ===
Research by Feigenson (2016) discusses the overall idea that juror emotions can affect guilt, blame and punishment. The core idea of the article follows the influence of integral and incidental emotions. Integral emotions are supposed to be caused by features of a particular case. Incidental emotions are emotions that arise, unrelated to the case. This foundational research posists that emotions may influence legal judgements through multiple different pathways {{expand}}.  

Rather than claiming that the research should be generalised to consider disgust, it should be considered that further research needs to be conducted to understand the role of moral disgust for jurors. This leaves open whether moral disgust causes jurors to be more certain or uncertain about information being presented.  

=== Moralisation and amplification hypotheses ===
A focus shift occured within some research frames in aiding to {{g}} understand the process of emotions for moral judgements. Jylkkä et al (2021) aimed to investigate whether visual media had a larger impact on disgust than gustatory stimuli. Researchers exposed {{who}} participants to visually disgusting or neutral pictures to know if participants could experience the intended emotion. 

Despite a clear activation of disgust, the results failed to support the theoretical predictions (Jylkkä et al, 2021). The moralisation hypothesis theorised that experiencing disgust will cause a person to find something generally neutral, to be morally wrong. This hypothesis was not supported. The amplification hypothesis in the study aimed to evaluate if experiencing disgusting and neutral videos made immoral actions seem to be significantly worse. It was also unsupported by the data (Jylkkä et al, 2021). Overall, within Jylkkä et al's {{g}} (2021) research, in these experiments visual disgust did not translate into harsher moral judgement. 

Bright and Goodman-Delahunty's (2006) research stands as a contrast to the impacts of disgust on moral judgement. Their research demonstrated that visceral disgust and anger motivated by gruesome crime scene photos significantly surged conviction rates. Images used in court that focus on victim harm may trigger socio-moral disgust (Bright and Goodman-Delahunty, 2006). Although Table 1 summarises findings driven primarily by anger rather than disgust, it gives context on how emotionally evocative crime scene evidence plays a role in sentencing.  

{| class=&quot;wikitable&quot;
|+Table 1: Summary of Bright &amp; Goodman-Delahunty (2006) {{g}} research
!Study
!Disgust trigger
!Core finding
|-
|Bright &amp; Goodman-Delahunty (2006)
|Case-specific crime scene photographs
|Anger increased punitiveness. 

* Conviction rate with gruesome crime scene photographs: 41.2%
* Conviction rate with neutral crime scene photographs: 38.2%
* Conviction rate without crime scene photographs: 8.8%
|}

==Conclusion==
The evidence that moral disgust influences juror's judgements of guilt, blame and punishment is mixed. The evidence mentioned prior reveals mixed results, some not even measuring disgust specifically. Going further, these theories and hypotheses should be utilised to target disgust, rather than assuming generalisability applies. Dual-process theory describes the process the brain may experience when determining guilt (Evans &amp; Stanovich, 2013). Under Heider's (1985) attribution theory, there is an expectation that juror's will logically balance personal causality to situational pressures to evaluate blame. There are disruptions to this as CCM reveals that spontaneous emotional reactions can prompt jurors to unconsciously blame the defendant (Alicke, 2000). Moral disgust can function through social categorisation cues as set out in the social identity theory (Tajfel &amp; Turner, 1979). Casting a defendant into the moral outgroup may lead to disgust providing an immediate psychological justification for harsher punishment. Practically, gruesome evidence may carry emotional weight (Bright &amp; Goodman-Delahunty, 2006) but jurors may be able to reconsider their initial reaction so emotional reactions do not determine a verdict (Feigenson, 2016).

This area of research could address gaps that would strengthen future studies. Given the mixed results from replication and standalone studies and meta-analytic work, further research should aim to establish how moral disgust works into the legal system. 

== See also ==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

[[w:Expectancy-value_theory|expectancy-value model]] 

https://en.wikipedia.org/wiki/Attribution_(psychology)

[[w:Dual_process_theory_(moral_psychology)|dual-process theory]] 

[[w:Disgust|Disgust]] 

==References==
{{Hanging indent|1=

Alicke, M. D. (2000). Culpable Control and the Psychology of Blame. Psychological Bulletin, 126(4), 556–574. https://doi.org/10.1037/0033-2909.126.4.556

Alicke, M. D., &amp; Weigel, S. H. (2021). The reasonable person standard: Psychological and legal perspectives. Annual Review of Law and Social Science, 17(1), 123–138. https://doi.org/10.1146/annurev-lawsocsci-111620-020400

Bornewasser, M., &amp; Bober, J. (1987). Individual, social group and intergroup behaviour. Some conceptual remarks on the social identity theory. European Journal of Social Psychology, 17(3), 267–276. https://doi.org/10.1002/ejsp.2420170303

Bright, D. A., &amp; Goodman-Delahunty, J. (2006). Gruesome evidence and emotion: Anger, blame, and jury decision-making. Law and Human Behavior, 30(2), 183–202. https://doi.org/10.1007/s10979-006-9027-y

Chapman, H. A., &amp; Anderson, A. K. (2012). Understanding disgust. Annals of the New York Academy of Sciences, 1251(1), 62–76. https://doi.org/10.1111/j.1749-6632.2011.06369.x 

Ouellet-Courtois, C., &amp; Radomsky, A. S. (2023). Can immorality be contracted? Appraisals of moral disgust and contamination fear. Behaviour Research and Therapy, 166, Article 104336. https://doi.org/10.1016/j.brat.2023.104336

DeSteno, D., Petty, R. E., Rucker, D. D., Wegener, D. T., &amp; Braverman, J. (2004). Discrete emotions and persuasion: The role of emotion-induced expectancies. Journal of Personality and Social Psychology, 86(1), 43–56. https://doi.org/10.1037/0022-3514.86.1.43

Ekman, P. (1992). Are there basic emotions? Psychological Review, 99(3), 550–553. https://doi.org/10.1037/0033-295X.99.3.550

Eskine, K. J., Kacinik, N. A., &amp; Prinz, J. J. (2011). A Bad Taste in the Mouth: Gustatory Disgust Influences Moral Judgment. Psychological Science, 22(3), 295–299. https://doi.org/10.1177/0956797611398497

Evans, J. St. B. T., &amp; Stanovich, K. E. (2013). Dual-Process Theories of Higher Cognition: Advancing the Debate. Perspectives on Psychological Science, 8(3), 223–241. https://doi.org/10.1177/1745691612460685

Feigenson, N. (2016). Jurors’ Emotions and Judgments of Legal Responsibility and Blame: What Does the Experimental Research Tell Us? Emotion Review, 8(1), 26-31. https://doi.org/10.1177/1754073915601223

Fishbein, M., &amp; Ajzen, I. (1975). Belief, attitude, intention, and behavior: an introduction to theory and research. Addison-Wesley Pub. Co.

Ghelfi E, Christopherson CD, Urry HL, et al. Reexamining the Effect of Gustatory Disgust on Moral Judgment: A Multilab Direct Replication of Eskine, Kacinik, and Prinz (2011). Advances in Methods and Practices in Psychological Science. 2020;3(1):3-23. doi:10.1177/2515245919881152

Hans, Valerie P. &quot;Jury systems around the world.&quot; Annual review of law and social science 4.1 (2008): 275-297. https://doi.org/10.1146/annurev.lawsocsci.4.110707.172319 

Heider, F. (1958). The psychology of interpersonal relations. John Wiley &amp; Sons. https://doi.org/10.1037/10628-000 

Jylkkä, J., Härkönen, J., &amp; Hyönä, J. (2021). Incidental disgust does not cause moral condemnation of neutral actions. Cognition &amp; emotion, 35(1), 96–109. https://doi.org/10.1080/02699931.2020.1810639

Rozin, P., &amp; Fallon, A. E. (1987). A perspective on disgust. Psychological Review, 94(1), 23–41. https://doi.org/10.1037/0033-295X.94.1.23

Rozin, P., Lowery, L., Imada, S., &amp; Haidt, J. (1999). The CAD triad hypothesis: A mapping between three moral emotions (contempt, anger, disgust) and three moral codes (community, autonomy, divinity). Journal of Personality and Social Psychology, 76(4), 574–586. https://doi.org/10.1037/0022-3514.76.4.574

Russell, P. S., &amp; Giner-Sorolla, R. (2013). Bodily moral disgust: What it is, how it is different from anger, and why it is an unreasoned emotion. Psychological Bulletin, 139(2), 328–351. https://doi.org/10.1037/a0029319

Salerno, J. M. (2021). The impact of experienced and expressed emotion on legal factfinding. Annual Review of Law and Social Science, 17(1), 181-203. https://doi.org/10.1146/annurev-lawsocsci-021721-072326

Salerno, J. M., &amp; Peter-Hagene, L. C. (2013). The Interactive Effect of Anger and Disgust on Moral Outrage and Judgments. Psychological Science, 24(10), 2069-2078. https://doi.org/10.1177/0956797613486988

Semmler, C., &amp; Brewer, N. (2002). Effects of mood and emotion on juror processing and judgments. Behavioral sciences &amp; the law, 20(4), 423–436. https://doi.org/10.1002/bsl.502

Tajfel, H., &amp; Turner, J. C. (1979). An integrative theory of intergroup conflict. In W. G. Austin &amp; S. Worchel (Eds.), The social psychology of intergroup relations (pp. 33–47).

}}

==External links==
* [https://www.youtube.com/watch?v=u-TmKo75gJI How Disgust Shapes our Thoughts on Moral Wrong &amp; the Political Right] (David Pizzaro)

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      <text bytes="26578" sha1="9lghqhw2qtvpyu2gbtyyjrwkozhphzt" xml:space="preserve">{{title|Dark empathy:&lt;br&gt;What is dark empathy, what are its consequences, and what can be done to address it?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
;Scenario

In a workplace, a new colleague, Addison, joined a close-knit team. He is charming and empathises with each team member. Addison fits in the team very well and many of the colleagues to him for advice. The lead role for a project becomes available and several members of the team decide to apply. One member, Charlie, takes Addison’s offer to help them prepare for the promotion. Charlie shares her fears and her perceived weaknesses for the role. Before the interviews, Addison strategically reminds Charlie about her weaknesses with a supportive twist. 

''“Good luck, Charlie. Take a breath and pause to avoid speaking too quickly. Despite your limited leadership experience, your skills speak for themselves. You got this!”'' 

Charlie felt very nervous and stumbled in the interview. She de-briefed with another colleague who said she had never heard her speak quickly when nervous. She was known for keeping a calm front under pressure. After a few weeks, the promotion is given to Addison. No one knew Addison had applied and the team was shocked since he had limited skills in this area of the work. 

{{RoundBoxBottom}}

Empathy can have a bright and dark side (Throop et al., 2020). The concern is when individuals with dark tetrad {{explain}} personality traits leverage their understanding of emotions and feelings to manipulate, control, or exploit to others.

Dark empathy is a developed skill that uses the understanding of emotion and people's vulnerabilities against them to control and manipulate for their advantage (Throop et al., 2020). To consider the use of dark empathy, we need understand why and how people use it. Individuals with high dark tetrads usually struggle to understand body language and emotion. The ability to understand emotion and feelings allows these dark trait individuals to go undetected for longer and have better control of those around them.

The ability to deceive and hide makes addressing the underlying behavioural problems harder. Their continued strategic and charming manipulation destroys the psychological safety of others around them. Research indicates that understanding emotional intelligence, the dark tetrad, and theory of mind could assist with understanding the use of dark empathy (Vonk et al., 2015).

{{RoundBoxTop|theme=1}}
;Focus questions

*How does dark empathy develop?  
*What drives an individual to use empathy to control and manipulate other people in their life?  
*How can psychological theory develop strategies to understand and assist individuals {{missing}} mitigate the negative effects of empathy? 

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== The development of empathy ==
[[Empathy]] is the ability to share feelings and thoughts of others while remaining self-other distinction (Håkansson Eklund &amp; Summer Meranius, 2021 {{ic|not in References}}). This means a person can understand someone else’s emotional experience while maintaining their identity. Empathy is not an innate but learned and developed {{aw}} {{f}}. It can be cultivated through techniques to understand other's feelings and foster [[wikipedia:Prosocial_behavior|pro-social behaviour]] (Abramson, 2021). Research found {{how}} that teenagers learn empathy through their parent’s {{g}} (Jiang et al., 2024), social groups (Decety, 2015), listening to music (Rabinowitch et al., 2013), and reading books particularly fiction (Bal &amp; Veltkamp, 2013).  Understanding how this foundational trait is normally developed provides a necessary baseline for examining how dark personalities can subvert empathy into a tool for manipulation.

=== The dark tetrad ===
[[File:Dark-triad-dark-tetrad-1.webp|thumb|'''Figure 1.''' The dark traits updated to include the fourth element of sadism. This created the dark tetrad of personality traits. ]]

Negative personality traits was {{g}} called the dark triad (Paulhus &amp; Williams, 2002). An expansion on these traits were to include a fourth element, sadism (Mededovic &amp; Petrovic, 2015). There are four elements to the dark tetrad (see Figure 1):
* Narcissism
* Machiavellianism
* Psychopathy
* Sadism

These traits demonstrate dark individuals have limited ability to comprehend other’s emotions and read their mental states (Vonk et al., 2015) {{g}}. {{what}} Individuals are self-centred, emotionally unstable, and focused on anti-social behaviour and this leads to few of them demonstrating empathy (Kaufman et al., 2019). See Table 1 for the breakdown of the {{what}} personality traits.

'''Table 1'''. Dark tetrad personality traits breakdown and behavioural signs {{f}}
{| class=&quot;wikitable&quot; style=&quot;margin: auto;&quot;
|-
! Personality traits !! Definition !! Behavioural signs
|-
| '''Narcissism''' || Self-importance and constant need for admiration || Arrogance, entitlement, and bragging
|-
| '''Machiavellianism''' || Strategic manipulation, calculated world view and general distrust of motives and integrity || Deception, exploitation, and emotional detachment
|-
| '''Psychopathy''' || Lack of empathy, poor impulse control, and fearlessness || Reckless, charming, no remorse, and cruel
|-
| '''Sadism''' || Deriving pleasure or enjoyment from harming others || Bullying, shaming, and humiliating others 
|}

=== Theory of mind ===
Theory of mind (ToM) is the ability to understand the internal mental states of others, including their beliefs and intentions (Wellman &amp; Liu, 2004). ToM plays an important role in understanding how dark trait individuals understand internal mental states to successfully navigate social environments. ToM and empathy are both important influences on [[Prosocial behavior|pro-social behaviour]] (Qiu et al., 2024), {{g}} which involves sharing, caring, helping, donating, comforting, and cooperating (Guo et al., 2023). While individuals with dark traits do not demonstrate these behaviours (Hurezan et al., 2026), ToM explains how dark empathy assists individuals to understand others and their own motives (Qiu et al., 2024). Dark trait individuals use this ability to read other's mental states and exploit it for their personal gain (Gojković et al., 2022). Recognising other’s vulnerablitlities while monitoring their own mental state, makes strategically exploiting easier. Ultimately, dark empaths use ToM to understand their own mental state to maintain their charming, empathic mask while concealing their manipulation of their victims.

=== Types of empathy ===
Empathy has bright and dark sides as well two types {{aw}} (Throop et al., 2020): 
* Affective empathy is physical and emotional sharing of feelings
**This involves experiencing with the other person which is found more commonly in bright empathy
* Cognitive empathy is the intellectual understanding of another’s emotions and thoughts without feeling their pain. 
**This involves understanding, yet keeping that self-other determination which makes it easier for manipulating by utilising emotions against another

Empathy relies on the individual to use it based on their morals and beliefs (Breithaupt &amp; Hamilton, 2019). A study on parent's {{g}} influence found {{how}} that empathy was not increased in teenagers with narcissistic traits (Jiang et al., 2024). This shows that dark traits {{g}} personalities are not affected by external influences. Therefore, understanding the specific type of empathy is key to understanding dark empathy. Cognitive empathy allows them to comprehend feelings and thoughts without experiencing the burden, {{g}} it suits strategic manipulation tactics like Machiavellianism (Gojković et al., 2022). When the motivation is to deceive and control, cognitive empathy provides the tools to serve their interests {{f}}. 

=== Making of a dark empath ===
Dark empathy is understanding those {{what}} feelings and harnessing them against others (Breithaupt &amp; Hamilton, 2019). It often develops when children experience trauma that forces them to predict and understand others' emotions just to stay safe (Bloxsom et al., 2021). Later in life, {{ic|people with}} dark traits (narcissism, Machiavellianism, psychopathy, and sadism) use this skill to deceive and control (Gojković et al.,2022).  Research assessing these personalities shows {{how}} that they do not share common societal ethics and morality (Mededovic &amp; Petrovic, 2015). Because they are less likely to be honest and fair, they use dark empathy and their flexible morals to justify manipulating vulnerabilities for their own advantage (Throop et al., 2020). This helps to achieve their goals. 

Dark empath {{sp}} is not a formal medical diagnosis. Instead, it is a term that mixes dark personality traits with high cognitive empathy (Duradoni et al., 2023). Cognitive empathy allows people to understand others’ thoughts and feelings without feeling them. This helps people with dark traits hide behind superficial charm, chronic lying, and a lack of remorse (Hepper et al., 2014; Vonk et al., 2015).  For example, narcissists can create a mask to hide their self-centred goals and better manipulate others (Hepper et al., 2014). Another sign is affective dissonance. This is a twisted emotional reaction, such as feeling happy when someone else suffers (Gojković et al., 2022). In the end, people with high dark traits may develop dark empathy to serve their goals.

== Consequences of dark empathy ==
{{RoundBoxTop|theme=3}}
;Scenario continued 

Charlie was assigned to report to Addison on the project and found herself becoming anxious at their one-on-one performance meetings. Charlie would spend over an hour preparing for the meetings despite keeping meticulous progress notes. She feels exhausted after each interaction and starts doubting her skills.

Addison finds it hard to keep up with the workload. Other team members have complained that Addison has not been running the project well. Addison’s boss reviews the work and finds Addison has not been monitoring the project. Charlie’s work had been the only praise the boss had for the update. Addison takes credit for Charlie's work and claims he has been managing Charlie's failing performance. Addison is to provide an update on the project and Charlie's performance. 
{{RoundBoxBottom}}

Empathy can lead to depression (Yan et al., 2021) as well as severe emotional distress and burnout (Huang et al., 2025). An individual can use cognitive empathy to deceive, manipulate and exploit (Huang et al., 2025). Using dark traits like charm and chronic lying (Gojković et al., 2022), dark empaths manipulate others to reach their goals while pretending to be an ally. Both the person using dark empathy and their victims suffer harm to their wellbeing (Heym et al., 2021). Perceiving emotions, planning a response, and controlling reactions take considerable energy. There is a physical component in controlling their body language and expression to disarm others (Throop et al., 2020). While bright empathy brings people together. {{g}} dark empathy isolates individuals from others and communities (Throop et al., 2020).

=== Effects on victims of dark empathy ===
The immediate impacts on their {{who}} relationships are subtle and unlikely to be noticed {{f}}. Dark empaths use their charming demeanour and their strategic understanding of other’s emotions and feelings to appear kind and empathic {{f}}. The strategic manipulation relies heavily on ToM to better understand their victim's weakness in order to exploit and deceive them {{f}}. After gaslighting an individual, their victim will develop chronic anxiety, depression, lower self-esteem, emotional dependency (Yan et al., 2021), shame, and humiliation (Throop et al., 2020). The victim's mental wellbeing may become apparent first and impact all areas of their lives {{f}}. This can isolate {{g}} victim and their mental state can become easier for a dark empath to manipulate. Additionally, the dark trait personalities are often linked to aggressive behaviours which increases the likelihood of intimate partner violence (Galán et al., 2026). Dark empathy can lead to harmful outcomes for dark empath's victims that become clearer with more exposure to the skillset.

=== Outcome for dark empaths ===
An individual using dark empathy may find the indifference and emotional de-attachment to others useful for achieving their goals (Throop et al., 2020). The emotional distance can intervene serve as a protective factor or buffer to psychological distress for the dark empath (Heym et al., 2021). While empathy combined with self-judgement and/or over-thinking can cause emotional distress (Heym et al., 2021), an individual who has dark traits and ToM, using cognitive empathy (understanding but not sharing emotions), does not experience the same level of that distress. The ability to not feel the emotions makes them better at utilising other’s vulnerabilities without the consequences {{f}}. Though controlling others is effective at first, keeping up the mask leads to burnout (Throop et al., 2020). The thought of losing control can cause aggressive behaviour, driven by fear and shame (di Giacomo et al., 2023). While dark trait individuals are more self-centred and lack the pro-social behaviours, hiding their nature can lead to depression and a loss of identity (Qiu et al., 2024).

== Addressing dark empathy ==
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;Scenario continued 

After talking to another colleague, Charlie notices her mental health is declining. She reaches out to the Employee Assistance Program (EAP) for support about her stress. The counsellor asks Charlie to consider when the anxiety and lower self-esteem started. She starts to realise it links back to Addison. 

Addison continues to blame Charlie for the project's failures. Addison fails to develop the deliverables in time for the boss’ deadline and a meeting is set up with him up with an EAP counsellor. Addison talks about Charlie's 'failures' and the counsellor asks Addison to consider his behaviour and input into the situation. 
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Dark empathy is not easily recognised as it can lower the appearance of aggressive behaviours in individuals with dark personalities traits (Heym et al., 2021). For example; a narcissist or sociopath may use understanding of ToM to influence others while a dark empath uses their cognitive empathy to control and most likely enjoy other's vulnerable emotions (Heym et al., 2021). The impacts of dark empathy can be found in the relationships of the dark empath. Person-centred approach assists in recognising patterns, identifying behaviours and setting boundaries (Galán et al., 2026).

=== Recovering from a dark empath ===
After dealing with a dark empath, an individual may find it hard to notice the patterns. Individual {{g}} needs to set healthy emotional boundaries in this relationship which can be assisted by seeking therapy to identify patterns and how to address them. Person-centre framework {{g}} (also known as client-centred approach) (Rogers, 1951) can assist in developing strategies and understanding the patterns they experienced (Galán et al., 2026). As victim's of dark empathy can develop mental health disorders such as PTSD, it's important to understand the patterns of abuse that affected their lives. Roger's non-judgemental approach in therapy can assist them to feel safe when victim's may have become guarded and mistrusting. Cognitive behavioural therapy (CBT) would be effective for unlearning the patterns of abuse and understanding their trauma (Iverson et al., 2011) from the dark empath. Dark empathy has a negative affect on those being exploited and bright empathy towards them will assist in recovering from the patterns of abuse. 

=== Unravelling from dark empathy ===
For a dark empath, the first step is recognizing their controlling behaviour and how it harms everyone's well-being. Carl Rogers' approach helps people understand who they are and who they want to be. This self-awareness lets them accept their dark traits and build stronger social skills (Galán et al., 2026). Some therapies that would be appropriate and effective for dark empaths are:
*Dialectial behaviour therapy
**This therapy allows people to talk through and accept their intense feelings (di Giacomo et al., 2023).
*[[w:Cognitive_behavioral_therapy|Cognitive Behaviour Therapy]] (CBT)
**This therapy assists in addressing anti-social behaviours and fosters pro-social behaviours (Shukla &amp; Upadhyay, 2025)

The steps to acknowledging the behaviours and traits are not {{g}} easy process. Considering their dark traits' lack of self-judgement, sense of self-importance, enjoy in harming others, these dark trait individuals show little desire to go to therapy for the sake of others (di Giacomo et al., 2023) {{g}}. Their dark traits cause anti-social behaviours and a disregard the impact of their actions {{f}}. Their lack self-judgement prevents them seeing themselves as part of the problem {{f}}.

== Conclusion ==
Dark empathy is a demonstration of how learning can be adapted to suit one’s desires {{vague}}. While it {{what}} can develop through parental influence (Jiang et al., 2024), social experience (Decety, 2015), listening to music (Rabinowitch et al., 2013), and reading fiction (Bal &amp; Veltkamp, 2013), it depends on morality of the individual. Understanding dark empathy is crucial for identifying and treating people with dark personality traits. Rather than genuinely sharing feelings, dark empaths use cognitive empathy as a tool to trick, manipulate, and exploit others (Heym et al., 2021). While traits like psychopathy and sadism usually involve anti-social behaviour, a lack of empathy, and enjoy {{g}} others' pain (Mededovic &amp; Petrovic, 2015), dark empaths fake emotional awareness and pro-social behaviour (Hurezan et al., 2026). This makes them difficult to spot, leaving victims vulnerable to gaslighting and emotional abuse. Consequently, standard person-centred frameworks are limited in effectiveness because these individuals rarely self-identify or seek help.

{{RoundBoxTop|theme=1}}
;Key takeaway messages

*Empathy can be developed, yet how it is used depends on the person's personality and sense of morals
*Empathy has a bright and dark side. Dark empathy is a lesser {{g}} recognised and its impacts have a large effect on their community.
*ToM can help understand how cognitive empathy is used. This is important in considering people's motivations of using dark empathy.
*Further research on dark empathy and the dark tetrad's use of it will significantly improve identification and assisting in developing effective prevention strategies for their victims
{{RoundBoxBottom}}

==See also==
*[[Motivation and emotion/Book/2020/Dark triad personality and emotion|Dark triad personality and emotion]] (Book chapter, 2020) 
*[[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and antisocial behaviour]] (Book chapter, 2019)
*[[Motivation and emotion/Book/2021/Emotional intelligence and the dark triad|Emotional intelligence and the dark triad]] (Book chapter, 2021)
*[[wikipedia:Machiavellianism_(psychology)|Machiavellianism]] (Wikipedia) 
*[[wikipedia:Narcissism|Narcissism]] (Wikipedia)
*[[wikipedia:Psychopathy|Psychopathy]] (Wikipedia)
*[[wikipedia:Dark_triad|The Dark Triad]] (Wikipedia)

==References==
{{Hanging indent|1=

Abramson, A. (2021, November 1). Cultivating empathy. Monitor on Psychology, 52(8). https://www.apa.org/monitor/2021/11/feature-cultivating-empathy

Bal, P. M., &amp; Veltkamp, M. (2013). How Does Fiction Reading Influence Empathy? An Experimental Investigation on the Role of Emotional Transportation. PloS one, 8(1), e55341. https://doi.org/10.1371/journal.pone.0055341 

Bloxsom, C. A. J., Firth, J., Kibowski, F., Egan, V., Sumich, A. L., &amp; Heym, N. (2021). Dark shadow of the self: How the dark triad and empathy impact parental and intimate adult attachment relationships in women. Forensic Science International: Mind and Law, 2, 100045. https://doi.org/https://doi.org/10.1016/j.fsiml.2021.100045 

Breithaupt, F., &amp; Hamilton, A. B. B. (2019). The dark sides of empathy (1st edition. ed.). Cornell University Press. https://doi.org/10.7591/9781501735608 

Decety, J. (2015). The neural pathways, development and functions of empathy. Current opinion in behavioural sciences, 3, 1-6. https://doi.org/10.1016/j.cobeha.2014.12.001 

di Giacomo, E., Andreini, E., Lorusso, O., &amp; Clerici, M. (2023). The dark side of empathy in narcissistic personality disorder. Front Psychiatry, 14, 1074558. https://doi.org/10.3389/fpsyt.2023.1074558 

Duradoni, M., Gursesli, M. C., Fiorenza, M., Donati, A., &amp; Guazzini, A. (2023). Cognitive Empathy and the Dark Triad: A Literature Review. European Journal of Investigation in Health, Psychology and Education (EJIHPE), 13(11), 2642-2680. https://doi.org/10.3390/ejihpe13110184 

Galán, M., Gerber, N., Valls, L., Molina, S., &amp; Pineda, D. (2026). Dark Empaths and Intimate Partner Violence: A Latent Profile Analysis of Dark Tetrad Traits and Empathy Dimensions in Face-to-Face and Cyber Violence Perpetration. Behavioural sciences, 16(10), 1780. https://doi.org/10.3390/bs16101780 

Gojković, V., Dostanić, J. S., &amp; Đurić, V. (2022). Structure of darkness: The Dark Triad, the ‘Dark’ Empathy and the ‘Dark’ Narcissism. Primenjena psihologija (Online), 15(2), 237-268. https://doi.org/10.19090/pp.v15i2.2380 

Guo, Q., Li, S., Liang, J., Yu, X., &amp; Lv, Y. (2023). Mindfulness may be associated with less prosocial engagement among high intelligence individuals. Scientific reports, 13(1), 4208. https://doi.org/10.1038/s41598-023-31039-3 

Hepper, E. G., Hart, C. M., &amp; Sedikides, C. (2014). Moving Narcissus: Can Narcissists Be Empathic? Personality &amp; social psychology bulletin, 40(9), 1079-1091. https://doi.org/10.1177/0146167214535812 

Heym, N., Firth, J., Kibowski, F., Sumich, A., Egan, V., &amp; Bloxsom, C. A. (2019). Empathy at the heart of darkness: Empathy deficits that bind the dark triad and those that mediate indirect relational aggression. Frontiers in psychiatry, 10, 95. 

Heym, N., Kibowski, F., Bloxsom, C. A. J., Blanchard, A., Harper, A., Wallace, L., Firth, J., &amp; Sumich, A. (2021). The Dark Empath: Characterising dark traits in the presence of empathy. Personality and Individual Differences, 169, 110172-110172. https://doi.org/10.1016/j.paid.2020.110172 

Huang, C., Wu, Z., Sha, S., Liu, C., Yang, L., Jiang, P., Zhang, H., &amp; Yang, C. (2025). The Dark Side of Empathy: The Role of Excessive Affective Empathy in Mental Health Disorders. Biological psychiatry (1969), 98(5), 404-415. https://doi.org/10.1016/j.biopsych.2024.12.020 

Hurezan, L., Miu, A. C., Costea, A., &amp; Visu-Petra, L. (2026). Dark Personality, Empathy, and Prosocial Behaviour in Convicted Offenders. Scientific reports, 16(1), 4117-4177. https://doi.org/10.1038/s41598-025-34229-3 

Iverson, K. M., Gradus, J. L., Resick, P. A., Suvak, M. K., Smith, K. F., &amp; Monson, C. M. (2011). Cognitive-Behavioural Therapy for PTSD and Depression Symptoms Reduces Risk for Future Intimate Partner Violence Among Interpersonal Trauma Survivors. Journal of consulting and clinical psychology, 79(2), 193-202. https://doi.org/10.1037/a0022512 

Jiang, X., Liu, C., Xiao, X., &amp; Zhan, Y. (2024). Role of narcissism and empathy in the relationship between parental attachment and moral response patterns. European journal of developmental psychology, 21(3), 365-387. https://doi.org/10.1080/17405629.2024.2325173 

Jonason, P. K., &amp; Kroll, C. H. (2015). A Multidimensional View of the Relationship Between Empathy and the Dark Triad. Journal of individual differences, 36(3), 150-156. https://doi.org/10.1027/1614-0001/a000166 

Kaufman, S. B., Yaden, D. B., Hyde, E., &amp; Tsukayama, E. (2019). The Light vs. Dark Triad of Personality: Contrasting Two Very Different Profiles of Human Nature [Original Research]. Frontiers in psychology, Volume 10 - 2019. https://doi.org/10.3389/fpsyg.2019.00467 

Mededovic, J., &amp; Petrovic, B. (2015). The Dark Tetrad: Structural Properties and Location in the Personality Space. Journal of individual differences, 36(4), 228-236. https://doi.org/10.1027/1614-0001/a000179 

Qiu, X., Gao, M., Zhu, H., Li, W., &amp; Jiang, R. (2024). Theory of mind, empathy, and prosocial behaviour in children and adolescent: A meta-analysis. Current psychology (New Brunswick, N.J.), 43(22), 19690-19707. https://doi.org/10.1007/s12144-024-05762-7 

Rabinowitch, T.-C., Cross, I., &amp; Burnard, P. (2013). Long-term musical group interaction has a positive influence on empathy in children. Psychology of music, 41(4), 484-498. https://doi.org/10.1177/0305735612440609 

Rogers, C. R. (1951). Client-centered therapy; its current practice, implications, and theory. Houghton Mifflin. 

Shukla, M., &amp; Upadhyay, N. (2025). Cold hearts and dark minds: A systematic review and meta-analysis of empathy across dark triad personalities. Frontiers in psychiatry, 16, 1546917. https://doi.org/10.3389/fpsyt.2025.1546917
 
Throop, C. J., Zahavi, D., de Lucas Freitas, J., Groark, K. P., Maibom, H., Satne, G. L., Stueber, K. R., &amp; Trnka, S. (2020). Dark and Bright Empathy: Phenomenological and Anthropological Reflections. Current anthropology, 61(3), 283-303. https://doi.org/10.1086/708844 

Vonk, J., Zeigler-Hill, V., Ewing, D., Mercer, S., &amp; Noser, A. E. (2015). Mindreading in the dark: Dark personality features and theory of mind. Personality and Individual Differences, 87, 50-54. https://doi.org/10.1016/j.paid.2015.07.025 

Wellman, H. M., &amp; Liu, D. (2004). Scaling of Theory-of-Mind Tasks. Child development, 75(2), 523-541. https://doi.org/10.1111/j.1467-8624.2004.00691.x 

Yan, Z., Zeng, X., Su, J., &amp; Zhang, X. (2021). The dark side of empathy: Meta‐analysis evidence of the relationship between empathy and depression. PsyCh journal (Victoria, Australia), 10(5), 794-804. https://doi.org/10.1002/pchj.482 

}}

==External links==
* [https://www.sciencefocus.com/wellbeing/dark-empaths Dark Empaths] (Science Focus)
* [https://www.psychologytoday.com/us/basics/dark-tetrad Dark Tetrad] (Psychology Today)
* [https://www.theguardian.com/science/2024/nov/10/narcissists-only-more-devious-the-truth-about-dark-empaths 'Narcissists - only more devious': the truth about dark empaths] (the Guardian)

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__TOC__

==Overview==
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[[File:Wolf Creek NFH nursing home fishing event October 2024.png|thumb|190px|'''Figure 1:''' Residents of an aged care home enjoying an outside activity ]]

;Imagine this...

Imagine you are an elderly resident with dementia living in a psychogeriatric ward in the beautiful Dutch countryside. On the walk from your bedroom to the communal dining room, you pass many people; staff, other residents and resident’s family members, but you do not recognise any of them. You take your breakfast to a table overlooking the garden and imagine walking outside, feeling the sun and smelling the flowers. Despite your longing to enjoy the garden, whether you get outside today is not up to you. You have been told it is unsafe, and you may only go out when a staff member has time to take you. How would this scenario make you feel? Trapped? Unsafe?

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This scenario draws on experiences that residents of a Dutch psychogeriatric ward shared with researchers studying what gives residents a sense of freedom and safety. They identified that time in nature, knowing the staff and spending time with others all contributed positively to a sense of personal freedom and safety in the ward (Van Andel &amp; Holkenborg, 2024).

Scenarios like this are common within aged care homes for residents living with dementia. A review of 1,158 articles published between 2000 and 2018 found that residents felt a lack of freedom, belonging, and connection, with some residents even describing their living situation as “living in a prison camp” (Shiells et al., 2020, p. 1563). Many residents describe their well-being as low with reports of poor self-esteem, connection, value and independence (Shiells et al., 2020).

This chapter asks how self-determination theory (SDT) can help understand and support the psychological needs of autonomy, competence, and relatedness (ACR) in people with dementia. It begins with an overview of dementia and models of care, introduces SDT as a framework for understanding motivation and well-being, then examines how ACR can be supported for people living with dementia and highlights where the evidence is still limited.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is dementia, and what care options exist to support people living with it?
* What does SDT propose about human motivation and well-being, and why is it a useful framework for understanding dementia care?
* What interventions increase well-being for dementia patients?
* How does person-centred care impact psychological well-being?

{{RoundBoxBottom}}

== Understanding dementia and models of care ==
Dementia is a neurodegenerative disease (Cipriani et al., 2020). 57 million people across the globe live with dementia as of 2021, making it the seventh most common cause of death globally (World Health Organisation, 2026). Dementia disproportionately affects women as they are both more likely to get dementia and more likely to look after people with it (World Health Organisation, 2026). There are four types of dementia: Alzheimer’s disease, [https://www.dementia.org.au/about-dementia/vascular-dementia vascular dementia], frontotemporal dementia and dementia Lewy body (Maclin et al., 2019). Alzheimer’s is the most common type of dementia, occurring in 69% of the population over 75. Alzheimer’s involves gradual memory and cognitive loss that impairs daily activities. The second most common is frontotemporal dementia and is distinguished by a distorted personality, along with loss of social and executive function, including in some cases the loss of speech (Maclin et al., 2019). Vascular dementia affects cognition from blood vessels in the brain becoming blocked resulting in deficits in the sensory and motor systems. Lewy bodies is characterised by [[wikipedia:Lewy_body|cellular inclusions]] in the brain that cause fluctuations in [[Cognition|cognitive function]], hallucinations and behavioural problems. Support is often needed for people with dementia in the early stages for complex daily activities such as organising transportation. As the disease progresses, help with basic activities like getting dressed is needed too (Cipriani et al., 2020).

Providing care and support for people living with dementia comes in many forms including in-home and institutionalised care. Families and patients choose between models based on a number of factors, including cultural values, knowledge of dementia, access to services, financial position and geographic location (Singh et al., 2014; Giebel et al., 2021). People receiving in-home care may be looked after by informal caregivers such as family members, formal caregivers or a mix of both, who are able to provide services at the home of the patient (Singh et al., 2014). Institutionalised care refers to the model of care where patients move into a facility where they can be monitored and looked after entirely by professionals or formal caregivers. In Australia, most people living with dementia choose in-home care. However, there are still over one million people living in institutionalised settings (Singh et al., 2014). 

== Self-determination theory ==
Self-determination theory (SDT) is a theory of motivation. SDT predicts motivation by explaining why people are intrinsically or extrinsically motivated to perform a behaviour (Dombestein et at, 2020; Ryan &amp; Deci, 2020).

If someone is intrinsically motivated to perform a behaviour it means they find motivation from internal factors with no influence from the environment. This could be learning to [[Portal:Dance|dance]] purely for the enjoyment of it. On the other hand, extrinsic motivation is when someone behaves in response to environmental factors (Ryan &amp; Deci, 2020). This could be learning to dance because all of their friends are dancing. People are very rarely motivated entirely for internal or external reasons therefore, extrinsic motivation is considered to be a continuum, starting with external regulation and finishing at integration:

* External regulation - behaviours are regulated from environmental cues, rewards and punishments.
* Introjection - environmental cues are being injected internally for instance the “punishment” of doing something wrong is feeling guilty.
* Identified - you identify with your behaviours for example you identify with being a good student, so you show up to class but do not enjoy class.
* Integration - there is less conflict between your identity and your behaviour. For example, you show up to class because you value being a good student and think it’s the right thing to do (Dombestein et al, 2020).

Motivation is multidimensional, as people often have several forms of motivation for the same task (Ryan &amp; Deci, 2020). For example, someone might run because their doctor advised it (identified) and because they enjoy seeing friends there (intrinsic). SDT offers an explanation as to where on the motivation continuum people operate. SDT says that for people to be intrinsically motivated or on the upper end of the extrinsic motivation scale you must satisfy all three critical psychological needs. These needs are:

[[File:Self-determination_theory_diagrame.png|thumb|347x347px|'''Figure 2.''' Self-determination theory diagram]]

* Autonomy - feeling ownership over actions and life circumstances
* Competence - a sense of confidence or mastery over performing tasks
* Relatedness - the need to feel like you belong to a group, that you are supported by others and in return people feel supported by you.

Achievement of these three psychological needs is not automatic, and people often require support to achieve these needs {{f}}. Organismic SDT states that people have a natural drive towards obtaining the critical psychological needs (Ryan &amp; Deci, 2020). Zou et al. (2026) found that Chinese girls chose menstrual products that increased bodily autonomy, confidence in areas they identified with, and closeness to family and media figures, but could only do so if they could afford them {{ic|how is this related to dementia care?}}. This reflects the theory that people have a natural desire to achieve the three psychological needs but need support from their environment.

There is some debate in motivation research as to whether the different types of motivation describe distinct qualitative forms of motivation or if they point to different quantitative points along a scale of motivation (Van den Broeck et al., 2021). SDT theorists view motivation as a continuum, supported by findings that students with higher ACR have better wellbeing, higher grades and more intrinsic motivation. (Bureau et al., 2022; Ryan &amp; Deci, 2020). 

{{RoundBoxTop|theme=3}}
'''Key points'''

* SDT is a theory of motivation. Extrinsic motivation sits on a continuum: external regulation, introjection, identified and integration.
* Autonomy, competence and relatedness must all be met for people to be intrinsically motivated or at the upper end of the continuum.
* People have a natural drive towards these needs but require support from their environment.
* Higher need satisfaction is linked to better wellbeing.

{{RoundBoxBottom}}

== Putting self-determination theory into practice: Meeting Psychological Needs in Dementia Patients ==
SDT proposes that supporting ACR in people living with dementia leads to better well-being, but these needs cannot just be given directly to a person {{f}}. Therefore, the strategies discussed below are best understood as shaping conditions rather than delivering outcomes. One of the most well-documented examples of SDT in practice is the process of facilitating patient decision-making. Colclough et al. (2026) found that it {{what}} did not improve well-being directly but did so by separately increasing satisfaction levels of all three needs, which is consistent with SDT’s claim that need satisfaction underlies well-being. However, decision-making is not the only pathway to increase ACR.

=== '''Competency''' ===
Competency has been seen {{aw}} to be supported by carers assisting or allowing patients where possible to perform tasks independently (Custers et al., 2011). When adults with dementia cannot perform simple tasks by themselves like eating, they report feeling ashamed and embarrassed (Estrada, 2021). The ability to perform tasks independently is one reason most people prefer to stay in their home (Wammes et al., 2021).

Both formal and informal caregivers help people with dementia feel competent and safe in their own home with patients reporting that their informal caregivers help them with daily living tasks, navigating the health system and organising social activities (Wammes et al., 2021). Similar sentiments have been shared around formal caregivers helping with medical information and daily tasks (Hoel et al., 2021). Aides for everyday activities are also cited as supporting competence (Rapaport et al., 2020; Wammes et al., 2021). The importance of the environment for allowing psychological needs to flourish is evident in respect to competency as people living in aged care homes are less likely to complete tasks independently as cognitive decline progresses, compared to people receiving informal care (Anselmo et al., 2025). Residents often report having little control over daily routines, such as what they eat or when they shower (Shiells et al., 2020) {{ic|isn't this about autonomy?}}. Assisting dementia patients to identify personal strengths and meaningful activities have been seen {{aw}} to increase self-direction (Döpp et al., 2021). Although the correlations are unclear, these studies suggest that helping residents identify personal strengths and meaningful activities could show caregivers where to allow more independence, such as control over daily routines {{ic|isn't this about autonomy?}}. 

=== Autonomy ===
Supporting autonomy in people with dementia is best understood through SDT’s definition of autonomy which is having a sense of agency over one’s life rather than the physical ability to act independently (Ryan &amp; Deci, 2020). A study by Stumpf et al. (2025) found {{how}} that people with dementia did not mind if they did not contribute much to decision making, as long as they were included in the process. 90% of people living with mild forms of dementia can articulate preferences and provide reasons for their preferences based on their values (Stumpf et al., 2025). Despite this, healthcare professionals will often defer all questions to patient’s family members (Stumpf et al., 2025). People with dementia generally accept the gradual loss of independence that comes with the progression of the disease if the decisions they do make around everyday things like going for a walk or dyeing their hair are respected (Estrada, 2021). Interventions that help to increase decision-making capacity by applying advance care planning to everyday situations, simplifying and visualising options, and staff knowing the residents have all helped increase autonomy (Van Der Weide et al., 2023). Van Der Weide et al. (2023) warned that offering fewer choices increases the risk of staff accidently letting their preferences dictate the options available to residents. They also found {{how}} that autonomy interventions only appeared valuable within trusting relationships with family and staff (Van Der Weide et al., 2023) {{expand}}.

=== Relatedness ===
The need for relatedness is often clearly expressed by people living in aged care homes with many residents reporting love and connection as important values (Van Andel &amp; Holkenborg, 2024). Shiells et al. (2020) {{how}} discovered that reading groups, exercising, and helping other residents with problems have all been linked {{ic|how strongly?}} to feeling valued. The provision of in-home care by formal caregivers has been seen to increase feelings of relatedness, as people report appreciating the conversation and liking when they had regular contact with the same carer, as that carer {{g}} better understood their preferences (Hoel et al., 2021). Lastly, decision-making has also been seen {{how}} to increase relatedness if ideas are heard and respected (Stumpf et al., 2025). Estrada (2021) offers a richer understanding as to why decision-making might increase relatedness through the principles of [https://pmc.ncbi.nlm.nih.gov/articles/PMC6465833/ person-centred care], by which you can develop a deeper connection through conversation. Many of these strategies depend on carers having the time, energy and support to deliver them. Therefore, person-centred care has also been explored as an intervention to help increase the well-being of carers.  

== Person-centred care impacts on promoting psychological well-being ==
Person- or client-centred care is an approach to care that focuses on maintaining a person’s sense of self by putting them and their needs at the forefront of importance when delivering care (Serbser-Koal et al., 2024). Person-centred care has been investigated for solutions to increase staff satisfaction in aged care homes and in-home services and to help to increase well-being in people with dementia. One of the principles of person-centred care is to ‘enter the person’s world’ through communication to develop a better connection (Serbser-Koal et al., 2024). Estrada (2021) revealed that one way you can increase connection with dementia patients is by simply allowing more time for conversations. People with dementia need more time to communicate because as their disease progresses their processing speed and [https://pubmed.ncbi.nlm.nih.gov/19672843/ short-term memory] capacity decreases (Estrada, 2021). Allowing enough time for communicating is important as many people express frustrations when they feel rushed (Estrada, 2021).

Even in people with non-fluent dementia, communication can increase relatedness through the encouragement of non-verbal communication like smiling, waving or touch (Estrada, 2021). Although it is well established that increased and better-quality communication between nursing, home staff and residents is needed, nursing staff report not having enough time to get to know residents (Van Andel &amp; Holkenborg, 2024) and therefore place more importance on basic survival needs over psychological needs (Estrada, 2021). In-home care staff have also reported that the lack of time to address patient needs is also a reason for high staff turnover (Lämås et al., 2021). Lämås et al. (2021) tested {{how}} person-centred care in home care services. 'Thriving' rose among people with dementia, but self-determination fell and formal caregivers did not benefit. However, the caregiver [https://pubmed.ncbi.nlm.nih.gov/19672843/ sample] was too {{how}} small to detect an effect, and the self-determination measure captured only autonomy, whereas the thriving measure (attitudes to home, care and relationships) likely reflected all three needs better. A study with a larger sample size {{ic|how large?}} by Vassbø et al. (2020) looked at the effects of a person-centred intervention over 14 months {{where}} and also found no effect on job satisfaction. Person-centred care therefore does not appear to be a solution to increase staff satisfaction. However, there are promising signs it may increase well-being in dementia patients.

== Increasing the well-being of informal caregivers ==
The psychological well-being of informal caregivers also affects the well-being of the people they care for. Informal caregivers report needing more emotional support, finding caring increasingly difficult as they become older, and report navigating the healthcare system on their own as challenging (Wammes et al., 2021). Interventions, based on SDT, that focus on increasing coping skills, communication skills and improving relationships made carers more intrinsically motivated to care for their loved ones, which led to increased wellbeing (Dombestein et al., 2020). Carers also reported that being able to drop their loved ones off at [https://agedcare.salvos.org.au/residential-aged-care/carpenter-court-aged-care-centre/ day-care centres] allowed them to socialise with others, thereby increasing relatedness, and provision of at-home aides that helped them complete domestic tasks helped them feel competent (Wammes et al., 2021).

==Conclusion==
Dementia makes independent living challenging, but it does not mean that agency should be taken away. Dementia patients are capable of decision-making even in severe cases. SDT seeks to explain well-being through the prism of intrinsic and extrinsic motivation. To become intrinsically motivated an individual must satisfy all three psychological needs. The strategies for increasing psychological well-being are most successful when the interconnected nature of the needs are understood (Slemp et al., 2024). Interventions aimed at one need often support others. For example, feeling genuinely listened-to and respected increases both relatedness and autonomy (Stumpf et al., 2025; Estrada, 2021). Because these needs overlap and cannot be supplied directly, the most effective interventions are those that change the environment to allow them to develop. Allowing more time to communicate, creating spaces where people can act more independently, and increasing collaboration in decision-making all support the satisfaction of one or more needs, which in turn supports the satisfaction of other needs. Carer well-being remains an important area for future research to help increase well-being for dementia patients. 

{{Robelbox|theme=13|icon=Nuvola_apps_korganizer.svg|iconwidth=48px|title=Quiz}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
&lt;quiz display=simple&gt;
{You operate a dementia care live-in clinic. Two of your employees come to you with competing ideas of how to make the lives of your patients better. Employee A suggests that you should provide the patients with a larger variety of food. Employee B suggests that you should allow patients to choose what they want to eat. Assuming you can implement either option, which would you choose?}

- A.	Larger variety of food 
+ B.	Choosing the weekly menu
&lt;/quiz&gt;
&lt;/div&gt; 
{{RoundBoxBottom}}

== See also ==
* [[Motivation and emotion/Book/2020/Basic psychological need theory|Basic psychological need theory]] (Book chapter, 2020)
* [[Caregiving and dementia]] (Wikiversity)
* [[Motivation and emotion/Book/2014/Dementia care motivation|Dementia care motivation]] (Book chapter, 2014)
* [[Dementia]] (Wikiversity)
* [[wikipedia:Self-determination theory|Self-determination theory]] (Wikipedia)
* [[Neurodegenerative disease]] (Wikiversity)

== References ==
{{Hanging indent|1=

Anselmo, A., Corallo, F., Pagano, M., Cardile, D., Marra, A., Maresca, G., De Luca, R., Alagna, A., Quartarone, A., Calabrò, R. S., &amp; Cappadona, I. (2025). Beyond diagnosis: Exploring residual autonomy in dementia through a systematic review. ''Medicina, 61''(5), Article 895. https://doi.org/10.3390/medicina61050895

Bureau, J. S., Howard, J. L., Chong, J. X. Y., &amp; Guay, F. (2022). Pathways to student motivation: A meta-analysis of antecedents of autonomous and controlled motivations. ''Review of Educational Research, 92''(1), 46–72. https://doi.org/10.3102/00346543211042426

Cipriani, G., Danti, S., Picchi, L., Nuti, A., &amp; Fiorino, M. D. (2020). Daily functioning and dementia. ''Dementia &amp; Neuropsychologia, 14''(2), 93–102. https://doi.org/10.1590/1980-57642020dn14-020001

Colclough, C., Perach, R., Harris, P., Rusted, J., Banerjee, S., &amp; Miles, E. (2026). Decision-making involvement and quality of life in people with dementia: The mediating role of psychological needs. ''Aging &amp; Mental Health, 30''(2), 267–276. https://doi.org/10.1080/13607863.2025.2541188

Custers, A. F. J., Kuin, Y., Riksen-Walraven, M., &amp; Westerhof, G. J. (2011). Need support and wellbeing during morning care activities: An observational study on resident–staff interaction in nursing homes. ''Ageing and Society, 31''(8), 1425–1442. https://doi.org/10.1017/S0144686X10001522

Dombestein, H., Norheim, A., &amp; Lunde Husebø, A. M. (2020). Understanding informal caregivers’ motivation from the perspective of self‐determination theory: An integrative review. ''Scandinavian Journal of Caring Sciences, 34''(2), 267–279. https://doi.org/10.1111/scs.12735

Döpp, C. M. E., Drenth, H., Verkade, P.-J., Francke, A. F., &amp; van der Heide, I. (2021). Interventions for improving self-direction in people with dementia: ''A systematic review. BMC Geriatrics, 21''(1), Article 195. https://doi.org/10.1186/s12877-021-02133-w

Estrada, R. (2021). Listen and you will see the person through the dementia. ''Journal for Person-Oriented Research, 7''(2), 88–97. https://doi.org/10.17505/jpor.2021.23891

Giebel, C., Robertson, S., Beaulen, A., Zwakhalen, S., Allen, D., &amp; Verbeek, H. (2021). “Nobody seems to know where to even turn to”: Barriers in accessing and utilising dementia care services in England and the Netherlands. International Journal of ''Environmental Research and Public Health, 18''(22), 12233. https://doi.org/10.3390/ijerph182212233

Hoel, K.-A., Rokstad, A. M. M., Feiring, I. H., Lichtwarck, B., Selbæk, G., &amp; Bergh, S. (2021). Person-centred dementia care in home care services – highly recommended but still challenging to obtain: A qualitative interview study. ''BMC Health Services Research, 21''(1), Article 723. https://doi.org/10.1186/s12913-021-06722-8

Lämås, K., Bölenius, K., Sandman, P.-O., Lindkvist, M., &amp; Edvardsson, D. (2021). Effects of a person-centred and health-promoting intervention in home care services– a non-randomized controlled trial. ''BMC Geriatrics, 21''(1), Article 720. https://doi.org/10.1186/s12877-021-02661-5

Maclin, J. M. A., Wang, T., &amp; Xiao, S. (2019). Biomarkers for the diagnosis of Alzheimer’s disease, dementia Lewy body, frontotemporal dementia and vascular dementia. ''General Psychiatry, 32''(1), e100054-n/a. https://doi.org/10.1136/gpsych-2019-100054

Rapaport, P., Burton, A., Leverton, M., Herat-Gunaratne, R., Beresford-Dent, J., Lord, K., Downs, M., Boex, S., Horsley, R., Giebel, C., &amp; Cooper, C. (2020). “I just keep thinking that I don’t want to rely on people.” a qualitative study of how people living with dementia achieve and maintain independence at home: Stakeholder perspectives. ''BMC Geriatrics, 20''(1), Article 5. https://doi.org/10.1186/s12877-019-1406-6

Ryan, R. M., &amp; Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', Article 101860. https://doi.org/10.1016/j.cedpsych.2020.101860

Serbser-Koal, J., Rommerskirch-Manietta, M., Purwins, D., &amp; Roes, M. (2024). Person-centredness in dementia care: An integrative review of theoretical approaches. ''BMJ Open, 14''(6), e085051. https://doi.org/10.1136/bmjopen-2024-085051

Shiells, K., Pivodic, L., Holmerová, I., &amp; Van den Block, L. (2020). Self-reported needs and experiences of people with dementia living in nursing homes: ''A scoping review. Aging &amp; Mental Health, 24''(10), 1553–1568. https://doi.org/10.1080/13607863.2019.1625303

Singh, P., Hussain, R., Khan, A., Irwin, L., &amp; Foskey, R. (2014). Dementia Care: Intersecting informal family care and formal care systems. ''Journal of Aging Research'', 109–117. https://doi.org/10.1155/2014/486521

Slemp, G. R., Field, J. G., Ryan, R. M., Forner, V. W., Van den Broeck, A., &amp; Lewis, K. J. (2024). Interpersonal supports for basic psychological needs and their relations with motivation, well-being, and performance: A meta-analysis. Journal of Personality and ''Social Psychology, 127''(5), 1012–1037. https://doi.org/10.1037/pspi0000459

Stumpf, A. D. H., McKenzie, E., &amp; Beavers, A. (2025). An account of personal autonomy for people living with dementia. ''The Gerontologist, 65''(11), 1. https://doi.org/10.1093/geront/gnaf210

Van Andel, S., &amp; Holkenborg, A. (2024). From the perspective of people with dementia: Using creative qualitative measures to assess the values and opinions on freedom and safety among people living with dementia. ''Healthcare,12''(14), 1412. https://doi.org/10.3390/healthcare12141412

Van Den Broeck, A., Howard, J. L., Van Vaerenbergh, Y., Leroy, H., &amp; Gagné, M. (2021). Beyond intrinsic and extrinsic motivation: A meta-analysis on self-determination theory’s multidimensional conceptualization of work motivation. ''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173

Van Der Weide, H., Lovink, M. H., Luijkx, K. G., &amp; Gerritsen, D. L. (2023). Supporting autonomy for people with dementia living in nursing homes: A rapid realist review. ''International Journal of Nursing Studies'',137, Article 104382. https://doi.org/10.1016/j.ijnurstu.2022.104382

Vassbø, T. K., Bergland, Å., Kirkevold, M., Lindkvist, M., Lood, Q., Sandman, P., Sjögren, K., &amp; Edvardsson, D. (2020). Effects of a person‐centred and thriving‐promoting intervention on nursing home staff job satisfaction: A multi‐centre, non‐equivalent controlled before–after study. ''Nursing Open, 7''(6), 1787–1797. https://doi.org/10.1002/nop2.565

Wammes, J. D., Labrie, N. H. M., Agogo, G. O., Monin, J. K., Bekker‐Grob, E. W., &amp; MacNeil Vroomen, J. L. (2021). Persons with dementia and informal caregivers prioritizing care: A mixed‐methods study. Alzheimer’s &amp; Dementia : ''Translational Research &amp; Clinical Interventions, 7''(1), Article e12193. https://doi.org/10.1002/trc2.12193

World Health Organisation. (2026). Dementia. https://www.who.int/news-room/fact-sheets/detail/dementia

Zou, W., Meadows, C. C. Z., Huang, L., &amp; Wang, B. (2026). Psychological determinants of menstrual product preferences: Applying self-determination theory to women’s reproductive health decision-making. J''ournal of Health Psychology, 31''(9), 3512–3527. https://doi.org/10.1177/13591053251407843

}}

== External links ==
* [https://www.ncbi.nlm.nih.gov/books/NBK551552/ Behavioural and psychological symptoms in dementia] (National library of medicine)
* [https://newsroom.ucla.edu/releases/dementia-care-study-highlights-importance-of-caregiver-self-efficacy Dementia care study highlights importance of caregiver self-efficacy] (UCLA newsroom)
* [https://findanexpert.unimelb.edu.au/scholarlywork/1753807-older-adults-using-technology-for-meaningful-activities-during-covid-19--an-analysis-through-the-lens-of-self-determination-theory Older] [https://findanexpert.unimelb.edu.au/scholarlywork/1753807-older-adults-using-technology-for-meaningful-activities-during-covid-19--an-analysis-through-the-lens-of-self-determination-theory adults using technology for meaningful activities during COVID-19: an analysis through the lens of self-determination theory] (University of Melbourne)

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__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Wolf Creek NFH nursing home fishing event October 2024.png|thumb|190px|'''Figure 1:''' Residents of an aged care home enjoying an outside activity ]]

;Imagine this...

Imagine you are an elderly resident with dementia living in a psychogeriatric ward in the beautiful Dutch countryside. On the walk from your bedroom to the communal dining room, you pass many people; staff, other residents and resident’s family members, but you do not recognise any of them. You take your breakfast to a table overlooking the garden and imagine walking outside, feeling the sun and smelling the flowers. Despite your longing to enjoy the garden, whether you get outside today is not up to you. You have been told it is unsafe, and you may only go out when a staff member has time to take you. How would this scenario make you feel? Trapped? Unsafe?

{{RoundBoxBottom}}

This scenario draws on experiences that residents of a Dutch psychogeriatric ward shared with researchers studying what gives residents a sense of freedom and safety. They identified that time in nature, knowing the staff and spending time with others all contributed positively to a sense of personal freedom and safety in the ward (Van Andel &amp; Holkenborg, 2024).

Scenarios like this are common within aged care homes for residents living with dementia. A review of 1,158 articles published between 2000 and 2018 found that residents felt a lack of freedom, belonging, and connection, with some residents even describing their living situation as “living in a prison camp” (Shiells et al., 2020, p. 1563). Many residents describe their well-being as low with reports of poor self-esteem, connection, value and independence (Shiells et al., 2020).

This chapter asks how self-determination theory (SDT) can help understand and support the psychological needs of autonomy, competence, and relatedness (ACR) in people with dementia. It begins with an overview of dementia and models of care, introduces SDT as a framework for understanding motivation and well-being, then examines how ACR can be supported for people living with dementia and highlights where the evidence is still limited.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is dementia, and what care options exist to support people living with it?
* What does SDT propose about human motivation and well-being, and why is it a useful framework for understanding dementia care?
* What interventions increase well-being for dementia patients?
* How does person-centred care impact psychological well-being?

{{RoundBoxBottom}}

== Understanding dementia and models of care ==
Dementia is a neurodegenerative disease (Cipriani et al., 2020). 57 million people across the globe live with dementia as of 2021, making it the seventh most common cause of death globally (World Health Organisation, 2026). Dementia disproportionately affects women as they are both more likely to get dementia and more likely to look after people with it (World Health Organisation, 2026). There are four types of dementia: Alzheimer’s disease, [https://www.dementia.org.au/about-dementia/vascular-dementia vascular dementia], frontotemporal dementia and dementia Lewy body (Maclin et al., 2019). Alzheimer’s is the most common type of dementia, occurring in 69% of the population over 75. Alzheimer’s involves gradual memory and cognitive loss that impairs daily activities. The second most common is frontotemporal dementia and is distinguished by a distorted personality, along with loss of social and executive function, including in some cases the loss of speech (Maclin et al., 2019). Vascular dementia affects cognition from blood vessels in the brain becoming blocked resulting in deficits in the sensory and motor systems. Lewy bodies is characterised by [[wikipedia:Lewy_body|cellular inclusions]] in the brain that cause fluctuations in [[Cognition|cognitive function]], hallucinations and behavioural problems. Support is often needed for people with dementia in the early stages for complex daily activities such as organising transportation. As the disease progresses, help with basic activities like getting dressed is needed too (Cipriani et al., 2020).

Providing care and support for people living with dementia comes in many forms including in-home and institutionalised care. Families and patients choose between models based on a number of factors, including cultural values, knowledge of dementia, access to services, financial position and geographic location (Singh et al., 2014; Giebel et al., 2021). People receiving in-home care may be looked after by informal caregivers such as family members, formal caregivers or a mix of both, who are able to provide services at the home of the patient (Singh et al., 2014). Institutionalised care refers to the model of care where patients move into a facility where they can be monitored and looked after entirely by professionals or formal caregivers. In Australia, most people living with dementia choose in-home care. However, there are still over one million people living in institutionalised settings (Singh et al., 2014). 

== Self-determination theory ==
Self-determination theory (SDT) is a theory of motivation. SDT predicts motivation by explaining why people are intrinsically or extrinsically motivated to perform a behaviour (Dombestein et at, 2020; Ryan &amp; Deci, 2020).

If someone is intrinsically motivated to perform a behaviour it means they find motivation from internal factors with no influence from the environment. This could be learning to [[Portal:Dance|dance]] purely for the enjoyment of it. On the other hand, extrinsic motivation is when someone behaves in response to environmental factors (Ryan &amp; Deci, 2020). This could be learning to dance because all of their friends are dancing. People are very rarely motivated entirely for internal or external reasons therefore, extrinsic motivation is considered to be a continuum, starting with external regulation and finishing at integration:

* External regulation - behaviours are regulated from environmental cues, rewards and punishments.
* Introjection - environmental cues are being injected internally for instance the “punishment” of doing something wrong is feeling guilty.
* Identified - you identify with your behaviours for example you identify with being a good student, so you show up to class but do not enjoy class.
* Integration - there is less conflict between your identity and your behaviour. For example, you show up to class because you value being a good student and think it’s the right thing to do (Dombestein et al, 2020).

Motivation is multidimensional, as people often have several forms of motivation for the same task (Ryan &amp; Deci, 2020). For example, someone might run because their doctor advised it (identified) and because they enjoy seeing friends there (intrinsic). SDT offers an explanation as to where on the motivation continuum people operate. SDT says that for people to be intrinsically motivated or on the upper end of the extrinsic motivation scale you must satisfy all three critical psychological needs. These needs are:

[[File:Self-determination theory diagram.png|thumb|347x347px|'''Figure 2.''' Self-determination theory diagram]]

* Autonomy - feeling ownership over actions and life circumstances
* Competence - a sense of confidence or mastery over performing tasks
* Relatedness - the need to feel like you belong to a group, that you are supported by others and in return people feel supported by you.

Achievement of these three psychological needs is not automatic, and people often require support to achieve these needs {{f}}. Organismic SDT states that people have a natural drive towards obtaining the critical psychological needs (Ryan &amp; Deci, 2020). Zou et al. (2026) found that Chinese girls chose menstrual products that increased bodily autonomy, confidence in areas they identified with, and closeness to family and media figures, but could only do so if they could afford them {{ic|how is this related to dementia care?}}. This reflects the theory that people have a natural desire to achieve the three psychological needs but need support from their environment.

There is some debate in motivation research as to whether the different types of motivation describe distinct qualitative forms of motivation or if they point to different quantitative points along a scale of motivation (Van den Broeck et al., 2021). SDT theorists view motivation as a continuum, supported by findings that students with higher ACR have better wellbeing, higher grades and more intrinsic motivation. (Bureau et al., 2022; Ryan &amp; Deci, 2020). 

{{RoundBoxTop|theme=3}}
'''Key points'''

* SDT is a theory of motivation. Extrinsic motivation sits on a continuum: external regulation, introjection, identified and integration.
* Autonomy, competence and relatedness must all be met for people to be intrinsically motivated or at the upper end of the continuum.
* People have a natural drive towards these needs but require support from their environment.
* Higher need satisfaction is linked to better wellbeing.

{{RoundBoxBottom}}

== Putting self-determination theory into practice: Meeting Psychological Needs in Dementia Patients ==
SDT proposes that supporting ACR in people living with dementia leads to better well-being, but these needs cannot just be given directly to a person {{f}}. Therefore, the strategies discussed below are best understood as shaping conditions rather than delivering outcomes. One of the most well-documented examples of SDT in practice is the process of facilitating patient decision-making. Colclough et al. (2026) found that it {{what}} did not improve well-being directly but did so by separately increasing satisfaction levels of all three needs, which is consistent with SDT’s claim that need satisfaction underlies well-being. However, decision-making is not the only pathway to increase ACR.

=== '''Competency''' ===
Competency has been seen {{aw}} to be supported by carers assisting or allowing patients where possible to perform tasks independently (Custers et al., 2011). When adults with dementia cannot perform simple tasks by themselves like eating, they report feeling ashamed and embarrassed (Estrada, 2021). The ability to perform tasks independently is one reason most people prefer to stay in their home (Wammes et al., 2021).

Both formal and informal caregivers help people with dementia feel competent and safe in their own home with patients reporting that their informal caregivers help them with daily living tasks, navigating the health system and organising social activities (Wammes et al., 2021). Similar sentiments have been shared around formal caregivers helping with medical information and daily tasks (Hoel et al., 2021). Aides for everyday activities are also cited as supporting competence (Rapaport et al., 2020; Wammes et al., 2021). The importance of the environment for allowing psychological needs to flourish is evident in respect to competency as people living in aged care homes are less likely to complete tasks independently as cognitive decline progresses, compared to people receiving informal care (Anselmo et al., 2025). Residents often report having little control over daily routines, such as what they eat or when they shower (Shiells et al., 2020) {{ic|isn't this about autonomy?}}. Assisting dementia patients to identify personal strengths and meaningful activities have been seen {{aw}} to increase self-direction (Döpp et al., 2021). Although the correlations are unclear, these studies suggest that helping residents identify personal strengths and meaningful activities could show caregivers where to allow more independence, such as control over daily routines {{ic|isn't this about autonomy?}}. 

=== Autonomy ===
Supporting autonomy in people with dementia is best understood through SDT’s definition of autonomy which is having a sense of agency over one’s life rather than the physical ability to act independently (Ryan &amp; Deci, 2020). A study by Stumpf et al. (2025) found {{how}} that people with dementia did not mind if they did not contribute much to decision making, as long as they were included in the process. 90% of people living with mild forms of dementia can articulate preferences and provide reasons for their preferences based on their values (Stumpf et al., 2025). Despite this, healthcare professionals will often defer all questions to patient’s family members (Stumpf et al., 2025). People with dementia generally accept the gradual loss of independence that comes with the progression of the disease if the decisions they do make around everyday things like going for a walk or dyeing their hair are respected (Estrada, 2021). Interventions that help to increase decision-making capacity by applying advance care planning to everyday situations, simplifying and visualising options, and staff knowing the residents have all helped increase autonomy (Van Der Weide et al., 2023). Van Der Weide et al. (2023) warned that offering fewer choices increases the risk of staff accidently letting their preferences dictate the options available to residents. They also found {{how}} that autonomy interventions only appeared valuable within trusting relationships with family and staff (Van Der Weide et al., 2023) {{expand}}.

=== Relatedness ===
The need for relatedness is often clearly expressed by people living in aged care homes with many residents reporting love and connection as important values (Van Andel &amp; Holkenborg, 2024). Shiells et al. (2020) {{how}} discovered that reading groups, exercising, and helping other residents with problems have all been linked {{ic|how strongly?}} to feeling valued. The provision of in-home care by formal caregivers has been seen to increase feelings of relatedness, as people report appreciating the conversation and liking when they had regular contact with the same carer, as that carer {{g}} better understood their preferences (Hoel et al., 2021). Lastly, decision-making has also been seen {{how}} to increase relatedness if ideas are heard and respected (Stumpf et al., 2025). Estrada (2021) offers a richer understanding as to why decision-making might increase relatedness through the principles of [https://pmc.ncbi.nlm.nih.gov/articles/PMC6465833/ person-centred care], by which you can develop a deeper connection through conversation. Many of these strategies depend on carers having the time, energy and support to deliver them. Therefore, person-centred care has also been explored as an intervention to help increase the well-being of carers.  

== Person-centred care impacts on promoting psychological well-being ==
Person- or client-centred care is an approach to care that focuses on maintaining a person’s sense of self by putting them and their needs at the forefront of importance when delivering care (Serbser-Koal et al., 2024). Person-centred care has been investigated for solutions to increase staff satisfaction in aged care homes and in-home services and to help to increase well-being in people with dementia. One of the principles of person-centred care is to ‘enter the person’s world’ through communication to develop a better connection (Serbser-Koal et al., 2024). Estrada (2021) revealed that one way you can increase connection with dementia patients is by simply allowing more time for conversations. People with dementia need more time to communicate because as their disease progresses their processing speed and [https://pubmed.ncbi.nlm.nih.gov/19672843/ short-term memory] capacity decreases (Estrada, 2021). Allowing enough time for communicating is important as many people express frustrations when they feel rushed (Estrada, 2021).

Even in people with non-fluent dementia, communication can increase relatedness through the encouragement of non-verbal communication like smiling, waving or touch (Estrada, 2021). Although it is well established that increased and better-quality communication between nursing, home staff and residents is needed, nursing staff report not having enough time to get to know residents (Van Andel &amp; Holkenborg, 2024) and therefore place more importance on basic survival needs over psychological needs (Estrada, 2021). In-home care staff have also reported that the lack of time to address patient needs is also a reason for high staff turnover (Lämås et al., 2021). Lämås et al. (2021) tested {{how}} person-centred care in home care services. 'Thriving' rose among people with dementia, but self-determination fell and formal caregivers did not benefit. However, the caregiver [https://pubmed.ncbi.nlm.nih.gov/19672843/ sample] was too {{how}} small to detect an effect, and the self-determination measure captured only autonomy, whereas the thriving measure (attitudes to home, care and relationships) likely reflected all three needs better. A study with a larger sample size {{ic|how large?}} by Vassbø et al. (2020) looked at the effects of a person-centred intervention over 14 months {{where}} and also found no effect on job satisfaction. Person-centred care therefore does not appear to be a solution to increase staff satisfaction. However, there are promising signs it may increase well-being in dementia patients.

== Increasing the well-being of informal caregivers ==
The psychological well-being of informal caregivers also affects the well-being of the people they care for. Informal caregivers report needing more emotional support, finding caring increasingly difficult as they become older, and report navigating the healthcare system on their own as challenging (Wammes et al., 2021). Interventions, based on SDT, that focus on increasing coping skills, communication skills and improving relationships made carers more intrinsically motivated to care for their loved ones, which led to increased wellbeing (Dombestein et al., 2020). Carers also reported that being able to drop their loved ones off at [https://agedcare.salvos.org.au/residential-aged-care/carpenter-court-aged-care-centre/ day-care centres] allowed them to socialise with others, thereby increasing relatedness, and provision of at-home aides that helped them complete domestic tasks helped them feel competent (Wammes et al., 2021).

==Conclusion==
Dementia makes independent living challenging, but it does not mean that agency should be taken away. Dementia patients are capable of decision-making even in severe cases. SDT seeks to explain well-being through the prism of intrinsic and extrinsic motivation. To become intrinsically motivated an individual must satisfy all three psychological needs. The strategies for increasing psychological well-being are most successful when the interconnected nature of the needs are understood (Slemp et al., 2024). Interventions aimed at one need often support others. For example, feeling genuinely listened-to and respected increases both relatedness and autonomy (Stumpf et al., 2025; Estrada, 2021). Because these needs overlap and cannot be supplied directly, the most effective interventions are those that change the environment to allow them to develop. Allowing more time to communicate, creating spaces where people can act more independently, and increasing collaboration in decision-making all support the satisfaction of one or more needs, which in turn supports the satisfaction of other needs. Carer well-being remains an important area for future research to help increase well-being for dementia patients. 

{{Robelbox|theme=13|icon=Nuvola_apps_korganizer.svg|iconwidth=48px|title=Quiz}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
&lt;quiz display=simple&gt;
{You operate a dementia care live-in clinic. Two of your employees come to you with competing ideas of how to make the lives of your patients better. Employee A suggests that you should provide the patients with a larger variety of food. Employee B suggests that you should allow patients to choose what they want to eat. Assuming you can implement either option, which would you choose?}

- A.	Larger variety of food 
+ B.	Choosing the weekly menu
&lt;/quiz&gt;
&lt;/div&gt; 
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== See also ==
* [[Motivation and emotion/Book/2020/Basic psychological need theory|Basic psychological need theory]] (Book chapter, 2020)
* [[Caregiving and dementia]] (Wikiversity)
* [[Motivation and emotion/Book/2014/Dementia care motivation|Dementia care motivation]] (Book chapter, 2014)
* [[Dementia]] (Wikiversity)
* [[wikipedia:Self-determination theory|Self-determination theory]] (Wikipedia)
* [[Neurodegenerative disease]] (Wikiversity)

== References ==
{{Hanging indent|1=

Anselmo, A., Corallo, F., Pagano, M., Cardile, D., Marra, A., Maresca, G., De Luca, R., Alagna, A., Quartarone, A., Calabrò, R. S., &amp; Cappadona, I. (2025). Beyond diagnosis: Exploring residual autonomy in dementia through a systematic review. ''Medicina, 61''(5), Article 895. https://doi.org/10.3390/medicina61050895

Bureau, J. S., Howard, J. L., Chong, J. X. Y., &amp; Guay, F. (2022). Pathways to student motivation: A meta-analysis of antecedents of autonomous and controlled motivations. ''Review of Educational Research, 92''(1), 46–72. https://doi.org/10.3102/00346543211042426

Cipriani, G., Danti, S., Picchi, L., Nuti, A., &amp; Fiorino, M. D. (2020). Daily functioning and dementia. ''Dementia &amp; Neuropsychologia, 14''(2), 93–102. https://doi.org/10.1590/1980-57642020dn14-020001

Colclough, C., Perach, R., Harris, P., Rusted, J., Banerjee, S., &amp; Miles, E. (2026). Decision-making involvement and quality of life in people with dementia: The mediating role of psychological needs. ''Aging &amp; Mental Health, 30''(2), 267–276. https://doi.org/10.1080/13607863.2025.2541188

Custers, A. F. J., Kuin, Y., Riksen-Walraven, M., &amp; Westerhof, G. J. (2011). Need support and wellbeing during morning care activities: An observational study on resident–staff interaction in nursing homes. ''Ageing and Society, 31''(8), 1425–1442. https://doi.org/10.1017/S0144686X10001522

Dombestein, H., Norheim, A., &amp; Lunde Husebø, A. M. (2020). Understanding informal caregivers’ motivation from the perspective of self‐determination theory: An integrative review. ''Scandinavian Journal of Caring Sciences, 34''(2), 267–279. https://doi.org/10.1111/scs.12735

Döpp, C. M. E., Drenth, H., Verkade, P.-J., Francke, A. F., &amp; van der Heide, I. (2021). Interventions for improving self-direction in people with dementia: ''A systematic review. BMC Geriatrics, 21''(1), Article 195. https://doi.org/10.1186/s12877-021-02133-w

Estrada, R. (2021). Listen and you will see the person through the dementia. ''Journal for Person-Oriented Research, 7''(2), 88–97. https://doi.org/10.17505/jpor.2021.23891

Giebel, C., Robertson, S., Beaulen, A., Zwakhalen, S., Allen, D., &amp; Verbeek, H. (2021). “Nobody seems to know where to even turn to”: Barriers in accessing and utilising dementia care services in England and the Netherlands. International Journal of ''Environmental Research and Public Health, 18''(22), 12233. https://doi.org/10.3390/ijerph182212233

Hoel, K.-A., Rokstad, A. M. M., Feiring, I. H., Lichtwarck, B., Selbæk, G., &amp; Bergh, S. (2021). Person-centred dementia care in home care services – highly recommended but still challenging to obtain: A qualitative interview study. ''BMC Health Services Research, 21''(1), Article 723. https://doi.org/10.1186/s12913-021-06722-8

Lämås, K., Bölenius, K., Sandman, P.-O., Lindkvist, M., &amp; Edvardsson, D. (2021). Effects of a person-centred and health-promoting intervention in home care services– a non-randomized controlled trial. ''BMC Geriatrics, 21''(1), Article 720. https://doi.org/10.1186/s12877-021-02661-5

Maclin, J. M. A., Wang, T., &amp; Xiao, S. (2019). Biomarkers for the diagnosis of Alzheimer’s disease, dementia Lewy body, frontotemporal dementia and vascular dementia. ''General Psychiatry, 32''(1), e100054-n/a. https://doi.org/10.1136/gpsych-2019-100054

Rapaport, P., Burton, A., Leverton, M., Herat-Gunaratne, R., Beresford-Dent, J., Lord, K., Downs, M., Boex, S., Horsley, R., Giebel, C., &amp; Cooper, C. (2020). “I just keep thinking that I don’t want to rely on people.” a qualitative study of how people living with dementia achieve and maintain independence at home: Stakeholder perspectives. ''BMC Geriatrics, 20''(1), Article 5. https://doi.org/10.1186/s12877-019-1406-6

Ryan, R. M., &amp; Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', Article 101860. https://doi.org/10.1016/j.cedpsych.2020.101860

Serbser-Koal, J., Rommerskirch-Manietta, M., Purwins, D., &amp; Roes, M. (2024). Person-centredness in dementia care: An integrative review of theoretical approaches. ''BMJ Open, 14''(6), e085051. https://doi.org/10.1136/bmjopen-2024-085051

Shiells, K., Pivodic, L., Holmerová, I., &amp; Van den Block, L. (2020). Self-reported needs and experiences of people with dementia living in nursing homes: ''A scoping review. Aging &amp; Mental Health, 24''(10), 1553–1568. https://doi.org/10.1080/13607863.2019.1625303

Singh, P., Hussain, R., Khan, A., Irwin, L., &amp; Foskey, R. (2014). Dementia Care: Intersecting informal family care and formal care systems. ''Journal of Aging Research'', 109–117. https://doi.org/10.1155/2014/486521

Slemp, G. R., Field, J. G., Ryan, R. M., Forner, V. W., Van den Broeck, A., &amp; Lewis, K. J. (2024). Interpersonal supports for basic psychological needs and their relations with motivation, well-being, and performance: A meta-analysis. Journal of Personality and ''Social Psychology, 127''(5), 1012–1037. https://doi.org/10.1037/pspi0000459

Stumpf, A. D. H., McKenzie, E., &amp; Beavers, A. (2025). An account of personal autonomy for people living with dementia. ''The Gerontologist, 65''(11), 1. https://doi.org/10.1093/geront/gnaf210

Van Andel, S., &amp; Holkenborg, A. (2024). From the perspective of people with dementia: Using creative qualitative measures to assess the values and opinions on freedom and safety among people living with dementia. ''Healthcare,12''(14), 1412. https://doi.org/10.3390/healthcare12141412

Van Den Broeck, A., Howard, J. L., Van Vaerenbergh, Y., Leroy, H., &amp; Gagné, M. (2021). Beyond intrinsic and extrinsic motivation: A meta-analysis on self-determination theory’s multidimensional conceptualization of work motivation. ''Organizational Psychology Review, 11''(3), 240–273. https://doi.org/10.1177/20413866211006173

Van Der Weide, H., Lovink, M. H., Luijkx, K. G., &amp; Gerritsen, D. L. (2023). Supporting autonomy for people with dementia living in nursing homes: A rapid realist review. ''International Journal of Nursing Studies'',137, Article 104382. https://doi.org/10.1016/j.ijnurstu.2022.104382

Vassbø, T. K., Bergland, Å., Kirkevold, M., Lindkvist, M., Lood, Q., Sandman, P., Sjögren, K., &amp; Edvardsson, D. (2020). Effects of a person‐centred and thriving‐promoting intervention on nursing home staff job satisfaction: A multi‐centre, non‐equivalent controlled before–after study. ''Nursing Open, 7''(6), 1787–1797. https://doi.org/10.1002/nop2.565

Wammes, J. D., Labrie, N. H. M., Agogo, G. O., Monin, J. K., Bekker‐Grob, E. W., &amp; MacNeil Vroomen, J. L. (2021). Persons with dementia and informal caregivers prioritizing care: A mixed‐methods study. Alzheimer’s &amp; Dementia : ''Translational Research &amp; Clinical Interventions, 7''(1), Article e12193. https://doi.org/10.1002/trc2.12193

World Health Organisation. (2026). Dementia. https://www.who.int/news-room/fact-sheets/detail/dementia

Zou, W., Meadows, C. C. Z., Huang, L., &amp; Wang, B. (2026). Psychological determinants of menstrual product preferences: Applying self-determination theory to women’s reproductive health decision-making. J''ournal of Health Psychology, 31''(9), 3512–3527. https://doi.org/10.1177/13591053251407843

}}

== External links ==
* [https://www.ncbi.nlm.nih.gov/books/NBK551552/ Behavioural and psychological symptoms in dementia] (National library of medicine)
* [https://newsroom.ucla.edu/releases/dementia-care-study-highlights-importance-of-caregiver-self-efficacy Dementia care study highlights importance of caregiver self-efficacy] (UCLA newsroom)
* [https://findanexpert.unimelb.edu.au/scholarlywork/1753807-older-adults-using-technology-for-meaningful-activities-during-covid-19--an-analysis-through-the-lens-of-self-determination-theory Older] [https://findanexpert.unimelb.edu.au/scholarlywork/1753807-older-adults-using-technology-for-meaningful-activities-during-covid-19--an-analysis-through-the-lens-of-self-determination-theory adults using technology for meaningful activities during COVID-19: an analysis through the lens of self-determination theory] (University of Melbourne)

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__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Sad Teen Girl Portrait.jpg|thumb|200x200px|'''Figure 1'''.  Even years after trauma, the fear of being hurt can remain]]

;Scenario

Imagine a girl named Zara, who experienced a traumatic event several years ago with an ex-boyfriend. This event is very much over, however, Zara continues to feel unsafe and insecure in relationships. When she meets a new boy she likes, she worries they are angry or that she's done something wrong, if they take a little while to respond to her messages. She often struggles to be vulnerable with these people and avoids communicating her feelings for fear of judgement and abandonment. Zara has recently made a new friend at college, someone who is incredibly kind, communicates openly, and doesn't make her second-guess. Despite these consistent, kind behaviours, Zara finds it difficult to believe that person will never hurt her.

{{ic|Consider adjusting the scenario to more clearly address the topic which is about rebuilding trust}}

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[[w:Trust (social science|Trust]] is crucial to the success, fulfilment, and longevity of romantic relationships. Trust isn't a single emotion or choice, but a vital concept built on intimacy, love, respect, and care. However, trust can be broken. In this chapter, we discuss how trauma survivors can develop their trust again in similar situations, while learning to accept the emotional responses that may be brought up from previous traumatic experiences. The following points explain the importance and psychological theories that can assist in understanding these processes.  

* A lack of trust negatively impacts a relationship, causing interpersonal problems through lying and anxiety
* Within monogamous relationships, emotional and physical infidelity is the most common experience that destroys trust with partners.
* Betrayal trauma leads to trauma responses, in many instances, causing great emotion and psychological damage.* Rebuilding trust is crucial to relationship recovery and health.
* Psychological science provides several perspectives that can explain the process of rebuilding trust in trauma survivors.
* This research and theory includes: Attachment theory and betrayal trauma theory

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;Focus questions

{{ic|Use bullet points as shown in Tutorial 2}}

1. How can experiences of betrayal trauma disrupt trust and influence future relationships?

2. Why do trauma survivors experience distrust in new situations that correspond with the original traumatic experience?

3. How can trauma survivors rebuild trust through new relationship experiences?
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== How Does Betrayal Trauma Affect Future Relationships? ==
Trauma can arise from an extremely distressing or disturbing experience and has lasting effects on an individual's psychological health (Das &amp; Jain, 2024). These traumatic experiences contribute to many psychological impacts, with symptoms of anxiety and depression, difficulties with trust, and disruptions to personal connections (Freyd et al., 2005). Betrayal trauma is the experience when a person or institution that an individual trusts violates that trust, leading to severe psychological harm (Freyd, 1994). When betrayal occurs within a romantic relationship, it can influence how they approach intimacy, trust, and reliability in future romantic relationships {{f}}.

=== Understanding Betrayal Trauma ===
Freyd's (1994) betrayal trauma theory suggests that the psychological impact of betrayal depends on the relationship between the person who experiences the harm and the person responsible for it. The theory suggests that when someone relies on a person who betrays them, it may be difficult to recognise the betrayal (Das &amp; Jain, 2024). Betrayal trauma assists in explaining the emotional hurt an individual is affected by after their trust and wellbeing are violated by significant people in their life (Freyd, 1994). Partner infidelity or “cheating” is a form of interpersonal trauma, which causes PTSD, depression and anxiety in 30% to 60% of betrayed people {{f}}. The act of intimate betrayal causes a traumatic memory. Clinically, this combination of symptoms points toward a stress-related disorder (Lonergan et al., 2020). An act of betrayal between romantic partners is one that harms a shared set of core beliefs established in the relationship and destroys the ability to have a partner as a source of safety and comfort (Lonergan et al., 2020). In romantic relationships, infidelity may challenge a victim's ability to trust their partner but also the expectations they established of honesty and reliability {{f}}. Understanding the psychological impacts of partner betrayal provides insights into how trust can be affected in future relationships.

=== Betrayal Trauma and Interpersonal Trust ===
According to the betrayal trauma theory {{f}}, individuals are susceptible to developing betrayal blindness when they are unable to fully recognise betrayal by someone they trust. In a relationship, betrayal blindness can be severely impactful {{f}}. Difficulty recognising betrayal can affect an individual's ability to judge someone's trustworthiness and leave them more vulnerable to being hurt in future relationships (Gobin &amp; Freyd, 2014). Zurbriggen and Freyd (2004) suggest that traumatic betrayals damage the cognitive functions that enable individuals to judge a person's trustworthiness, which could lead them to trust unworthy people and increase the chances of future betrayal. Supporting this theory, Gobin and Freyd (2009) discovered {{how}} that victims {{where}} of high betrayal trauma (HBT) had an increased likelihood of staying in a romantic relationship after experiencing infidelity and a betrayal of trust. 

Gobin and Freyd (2014) conducted an experiment investigating the relationship between betrayal trauma and interpersonal trust by examining their {{who}} participants' trust in others, their romantic partners, and their behaviours in an experimental trust game. They discovered that participants who reported HBT also reported having lower trust in romantic partners. Following infidelity, victims may be hesitant and show caution when trusting future partners, because they don't wish to experience betrayal again. However, developing trust may involve not only learning to trust romantic partners in future relationships but also regaining confidence in their own judgement of others.


Jennifer J. Freyd's Two-Dimensional Model for Traumatic Events

=== Attachment and Intimacy after Betrayal ===
Interpersonal trauma has been shown to influence attachment styles, with a dependency on the attachment between the victim and the perpetrator (Brennan et al., 1998, as cited in Das &amp; Jain, 2024). Although attachment theory was originally developed to explain relationships between children and their caregivers, researchers have since applied it to romantic relationships (Das &amp; Jain, 2024). Attachment theory emphasises human beings' natural tendency to form and maintain loving, caring, and affectionate bonds with others. (Bowlby, 1988 as cited in Mosley, 2022). In a romantic relationship, secure attachment is associated with emotional security, physical closeness, and a reciprocal sense of safety between partners. However, betrayal by a romantic partner threatens relationship safety and security, resulting in an attachment injury, a violation of trust that affects a victim's beliefs about the relationship (Johnson et al., 2001, as cited in Mosley, 2022). Research suggests that the more intimate the relationship in which the betrayal occurs, the more likely it is to create traumatic memories that affect a person's attachment (Butler et al., 2021, as cited in Mosley, 2022). This has devastating consequences, not only psychologically and physically, because a victim might also struggle to feel confident in another person's trustworthiness as a result. Further impacting how they approach trust and future romantic relationships.

== How Does Betrayal Trauma Affect Well-being? ==
Betrayal trauma impacts the psychological and physiological health of betrayal trauma victims by increasing fear and stress in future relationships. Research links betrayal trauma with difficulties in emotional regulation and mental illness symptoms associated with PTSD, depression and dissociation (Mosley, 2022; Ward, 2026).

=== Physical and Emotional Responses ===
In recent history, trauma responses performed by a victim have been associated with feelings of fear during and after the traumatic experience {{f}}. When we experience trauma, our body and brain's primary reaction is fear, triggering a range of physiological and emotional responses (DePrince et al., 2011). In the context of betrayal, the emotional reaction can be described as similar to the emotions people feel when confronted with death or horrific events (Mosley, 2022). Research suggests that emotional regulation capacities are impacted among survivors of Betrayal trauma, with great impacts towards psychological and physiological functioning (Mosley, 2022). Continuous exposure to relationship problems has been shown to disrupt stress regulation systems, with the prefrontal cortex and limbic reactivity most affected {{f}}. Evidence from trauma and stress research supports the likelihood of neurobiological dysregulation in individuals experiencing relationship betrayal (Ward, 2026). Clinical studies have further observed a change in emotional regulation and impaired executive functioning after continuous betrayal is experienced (Ward, 2026). There are also great physical consequences as a result of experiencing betrayal trauma. Our bodies' autonomic nervous system works within nanoseconds, activating to prepare for threats. Our bodies are built to detect danger by increasing adrenaline and cortisol, which prepare us to fight, flee, or shut down (Sutherland, 2019, as cited in Ward, 2026). However, the body works by continuously returning to equilibrium, responding rapidly to stress, and then relaxing {{f}}. Victims of betrayal, as a result of heightened stress, experience an unhealthy state of emotional threat and danger {{f}}. López Martínez et al. (2018, as cited in Ward, 2026) found that the body's continual response to betrayal, and a heightened sense of dread about future betrayals, creates emotional dysregulation by keeping the response system persistently engaged. Researchers have also discovered that victims of high betrayal trauma are more susceptible to experiencing pain in the form of soft tissue, chronic fatigue, diabetes, bladder issues, pelvic pain and asthma (Silber, 2020).

=== Post-traumatic stress disorder and Mental Illness ===
PTSD, or post-traumatic stress disorder, is a mental illness most noticeably caused by previous traumatic experiences, characterised by symptoms of depression, alcohol and drug abuse, dissociative tendencies, personality disorders and health concerns (Bremner, 2006). Betrayal trauma has been associated with multiple psychological diagnoses. These include PTSD, depression and dissociation (Mosley, 2022). Through a variety of studies, betrayal trauma was further associated with symptoms of PTSD, being impacted by intrusive thoughts and an impairment in self-concept (Ward, 2026). Atlas and Ingram (1998, as cited in Freyd et al., 2005) reported a stronger association between betrayal trauma and PTSD than between non-betrayal trauma and PTSD. Further research conducted by Kelley (2009, as cited in DePrince et al., 2011) discovered that insights of betrayal had a stronger association with PTSD than perceptions of life threat. Furthermore, Kaehler and Freyd (2009, DePrince et al., 2011) found an association between middle and high levels of betrayal trauma exposure and borderline personality characteristics among a college student population.

== How Can Trauma Survivors Rebuild Trust and Feel Safe in Future Relationships? ==
Rebuilding trust after infidelity is a gradual process which takes time and emphasises the importance for consistent honest, behaviour and support {{f}}. Counselling, forgiveness and managing painful memories can work towards rebuilding trust (Abrahamson et al., 2012). However recovery is dependant on the victims {{g}} decision about whether they will continue with the relationship. 

=== Rebuilding Trust and Emotional Safety ===
Although infidelity is a severely damaging event, not all couples choose to end their relationship after discovering an affair (Abrahamson et al., 2012). Trust in a relationship involves accepting a degree of vulnerability in relation to expectations of monogamy shaped by a partner's intentions and behaviour (Rousseau et al., 1998, as cited in Pramudito &amp; Minza, 2021). Infidelity damages these expectations, causing the betrayed partner to lose confidence in their partner's honesty and reliability (Pramudito &amp; Minza, 2021). Rebuilding trust is a gradual process that requires more than forgiveness. Olson, Russell, Higgins-Kessler, and Miller (2002, as cited in Abrahamson et al., 2012) describe a three-phase step to rebuilding trust. The first phase is known as the roller coaster phase, describing the continual change or “roller coaster” of emotions. This can be seen in the initial shock, strong feelings of disclosure, and the many uncontrollable emotions that come up. The second phase, known as the moratorium phase, begins when the couple agrees to set  hurt aside and move forward with their lives. Lastly, the third phase suggests the couple look to the future and focus on rebuilding trust in the relationship. Abrahamson et al. (2012) identified that forgiveness, counselling, managing painful memories of the infidelity, and changes in that relationship are all steps towards rebuilding that trust. In addition, Anderson (2025) suggested that viewing infidelity through a trauma-based perspective may assist marriage counsellors to better recognise and assist clients' experiences and provide further evidence-based support. Furthermore, Pramudito and Minza (2021) emphasised the importance of restoring trust over time through consistent demonstrations of faithfulness and honest behaviour. However, more importantly, the perspectives of the betrayed partner are necessary, as it is their decision to remain within the relationship in order to learn to rebuild trust again (Mitchell et al., 2025). However, Mitchell et al. (2025) explains that the betrayed partner may feel pressure to forgive in order for the relationship to continue, and this responsibility, along with overwhelming, ever-changing emotions, can affect their ability to make accurate, necessary decisions about the relationship. To rebuild the victim's trust in the perpetrator, signs of faithfulness and integrity are crucial. Goldie (2013, as cited in Pramudito and Minza, 2021). Furthermore, there is an emphasis on resolving conflicts positively in order to rebuild the victim's trust and to rebuild trustworthiness to work towards building a fully recovered marital relationship (Baucom et al., 2009, as cited in Pramudito &amp; Minza, 2021; Dickie, 2015, as cited in Pramudito &amp; Minza, 2021).

{{ic|Use the [[Help:Quiz]] quiz tool}}

'''According to the betrayal trauma theory, why can betrayal from a trusted partner make it more difficult in the future to trust another partner?'''

{| class=&quot;wikitable&quot;
|+
! colspan=&quot;3&quot; valign=&quot;top&quot; |Quiz 1 - Multiple Choice 
|-
! valign=&quot;top&quot; |A
| valign=&quot;top&quot; |It makes them dislike their parents and all their friends
| rowspan=&quot;4&quot; |Answer (B)
|-
! valign=&quot;top&quot; |B
| valign=&quot;top&quot; |It ruins their trust, safety, and judgement of whether others are trustworthy
|-
! valign=&quot;top&quot; |C
| valign=&quot;top&quot; |It makes them unable to communicate with people
|-
! valign=&quot;top&quot; |D
| valign=&quot;top&quot; |It makes them sad
|}

'''Zara was cheated on by her ex-boyfriend. It's a few years later now and she has a new boyfriend. They are both incredibly happy. However, as the days go by, and not to his fault, she becomes more noticeable of the small changes in her partners behaviours as they develop their relationship. One day he forgets to message her while he's at work. Even though it was a mistake she still feels like something is wrong. What is she experiencing?''' 

{| class=&quot;wikitable&quot;
|+
! colspan=&quot;3&quot; valign=&quot;top&quot; |'''Quiz 2 - Multiple Choice''' 
|-
! valign=&quot;top&quot; |A
| valign=&quot;top&quot; |Confidence, in herself and her boyfriend
| rowspan=&quot;4&quot; |Answer (C)
|-
! valign=&quot;top&quot; |B
| valign=&quot;top&quot; |Fatigue, she has had a very long day
|-
! valign=&quot;top&quot; |C
| valign=&quot;top&quot; |Hypervigilance, she is highly alert to supposed danger
|-
! valign=&quot;top&quot; |D
| valign=&quot;top&quot; |Brain fog, she forgets what has happened
|}

== Conclusion ==
Experiences of betrayal, especially those within romantic relationships, are can {{g}} be traumatic and have lasting impacts on a victim's sense of trust and emotional safety in the people around them. Betrayal trauma theory assists in explaining how harm caused by a trusted person can impact beliefs about relationships, while attachment theory assists us in understanding how these experiences may affect intimacy and emotional security. However, the difficulties victims face do not mean they are incapable of developing trust in relationships again. Rebuilding trust is a gradual process in which we learn to recognise signs of trustworthiness and develop confidence in our ability to judge others' trustworthiness. Relationships are built on communication, reliability, safety, and honesty, and experiencing this in a new relationship may build a greater sense of safety. Attempting to recover after infidelity will take time, trust will take time to fully restore, and forgiveness will continue to develop. However, rebuilding trust is not about guaranteeing that betrayal will ever happen again, but is about learning to recognise trustworthiness and to adopt a sense of security that will help victims form meaningful connections despite past hurt.

==See also==
Wikipedia Link - PTSD &lt;ref&gt;{{Cite journal|date=2026-09-05|title=Post-traumatic stress disorder in children and adolescents|url=https://en.wikipedia.org/w/index.php?title=Post-traumatic_stress_disorder_in_children_and_adolescents&amp;oldid=1373411231|journal=Wikipedia|language=en}}&lt;/ref&gt;
Discusses PTSD in children and adolescents and the diagnosis, risk factors, trauma and symptoms. [https://en.wikipedia.org/wiki/Post-traumatic_stress_disorder_in_children_and_adolescents]

Wikiversity Link – Trauma and Attachment Development &lt;ref&gt;{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2025/Trauma_and_attachment_development|title=Motivation and emotion/Book/2025/Trauma and attachment development - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-11}}&lt;/ref&gt;
Discusses various attachment styles and trauma and the way they both interconnect.[https://en.wikipedia.org/wiki/Post-traumatic_stress_disorder_in_children_and_adolescents]

==References==
{{Hanging indent|1=

Abrahamson, I., Hussain, R., Khan, A., &amp; Schofield, M. J. (2012, October 21). ''What helps couples rebuild their relationship after infidelity?'' SAGE Publications. ResearchGate. https://www.researchgate.net/publication/233341758_What_Helps_Couples_Rebuild_Their_Relationship_After_Infidelity

Affair Recovery. (2021, July 1). ''How to rebuild trust after infidelity: Become reliable'' [Video]. YouTube. https://www.youtube.com/watch?v=k8foEEWnsFw

Anderson, M. (2025). The impact and connection between infidelity and PTSD. ''Journal of Counseling Sexology and Sexual Wellness'', ''6''(2). [https://digitalcommons.unf.edu/jcssw/vol6/iss2/ https://doi.org/10.34296/06022017]

Bremner, J. D. (2006). Traumatic stress: Effects on the brain. ''Dialogues in Clinical Neuroscience'', ''8''(4), 445–461. [https://www.tandfonline.com/doi/full/10.31887/DCNS.2006.8.4/jbremner https://doi.org/10.31887/dcns.2006.8.4/jbremner]

Das, M., &amp; Jain, N. (2024). Partner Betrayal Trauma and trust: Understanding the impact on attachment style and self-esteem. ''European Journal of Trauma &amp; Dissociation'', ''8''(2), 100409–100409. [https://www.sciencedirect.com/science/article/abs/pii/S2468749924000334?via%3Dihub https://doi.org/10.1016/j.ejtd.2024.100409]

DePrince, A. P., Brown, L. S., Cheit, R. E., Freyd, J. J., Gold, S. N., Pezdek, K., &amp; Quina, K. (2011). Motivated forgetting and misremembering: Perspectives from betrayal trauma theory. ''Nebraska Symposium on Motivation'', 193–242. [https://link.springer.com/chapter/10.1007/978-1-4614-1195-6_7 https://doi.org/10.1007/978-1-4614-1195-6_7]

Ford, J. D, &amp; Courtois, C. A. (2016). ''Treating complex traumatic stress disorders in children and adolescents : Scientific foundations and therapeutic models''. New York ; London The Guilford Press. https://books.google.com.au/books?hl=en&amp;lr=&amp;id=Uv0E4W3bMYcC&amp;oi=fnd&amp;pg=PA62&amp;dq=betrayal+trauma+theory+1996&amp;ots=cKuxJAUZUq&amp;sig=-k98_BIIGp4gzPyPBcidj1WZ-qI#v=onepage&amp;q=betrayal%20trauma%20theory%201996&amp;f=false

Freyd, J. J. (1994). Betrayal trauma: Traumatic amnesia as an adaptive response to childhood abuse. ''Ethics &amp; Behavior'', ''4''(4), 307–329. [https://www.tandfonline.com/doi/abs/10.1207/s15327019eb0404_1 https://doi.org/10.1207/s15327019eb0404_1]

Freyd, J. J. (2013). ''Definition of Betrayal Trauma Theory''. Uoregon.Edu. https://dynamic.uoregon.edu/jjf/defineBT.html

Freyd, J. J., Klest, B., &amp; Allard, C. B. (2005). Betrayal trauma: Relationship to physical health, psychological distress, and a written disclosure intervention. ''Journal of Trauma &amp; Dissociation'', ''6''(3), 83–104. [https://www.tandfonline.com/doi/abs/10.1300/J229v06n03_04 https://doi.org/10.1300/j229v06n03_04]

Gobin, R. L., &amp; Freyd, J. J. (2009). Betrayal and revictimization: Preliminary findings. ''Psychological Trauma: Theory, Research, Practice, and Policy'', ''1''(3), 242–257. [https://psycnet.apa.org/doiLanding?doi=10.1037%2Fa0017469 https://doi.org/10.1037/a0017469]

Gobin, R. L., &amp; Freyd, J. J. (2014). The impact of betrayal trauma on the tendency to trust. ''Psychological Trauma: Theory, Research, Practice, and Policy'', ''6''(5), 505–511. [https://psycnet.apa.org/doiLanding?doi=10.1037%2Fa0032452 https://doi.org/10.1037/a0032452]

Lonergan, M., Brunet, A., Rivest‐Beauregard, M., &amp; Groleau, D. (2020). Is romantic partner betrayal a form of traumatic experience? A qualitative study. ''Stress and Health'', ''37''(1), 19–31. [https://onlinelibrary.wiley.com/doi/10.1002/smi.2968 https://doi.org/10.1002/smi.2968]

Martin, C. G., Cromer, L. D., DePrince, A. P., &amp; Freyd, J. J. (2013). The role of cumulative trauma, betrayal, and appraisals in understanding trauma symptomatology. ''Psychological Trauma: Theory, Research, Practice, and Policy'', ''5''(2), 110–118. https://doi.org/10.1037/a0025686

Mitchell, E. A., Brown, K. S., Spencer, J., &amp; Harris, K. (2025). Staying together after infidelity: An exploration of the decision-making process of recovery from the perspective of the injured partner. ''Journal of Marital and Family Therapy'', ''52''(1), e70110. PubMed Central. [https://onlinelibrary.wiley.com/doi/10.1111/jmft.70110 https://doi.org/10.1111/jmft.70110]

Mosley, T. L. (2022). ''Forever imprinted: Understanding the relationship between betrayal trauma within intimate relationships and attachment - ProQuest''. Proquest.Com. https://www.proquest.com/openview/c437e8a57b6e9a35babbdda9171ccee0/1?pq-origsite=gscholar&amp;cbl=18750&amp;diss=y

Pramudito, A. A., &amp; Minza, W. M. (2021). The dynamics of rebuilding trust and trustworthiness in marital relationship post infidelity disclosure. ''Jurnal Psikologi'', ''48''(2), 16. [https://eprints.ukmc.ac.id/6452/5/Published%20-%20The%20Dynamics%20of%20Rebuilding%20Trust%20and%20Trustworthiness%20in%20Marital%20Relationship%20Post%20Infidelity%20Disclosure.pdf https://doi.org/10.22146/jpsi.60974]

Silber, D. (2020). ''Trust again'' (pp. 1–229). Rowman &amp; Littlefield Publishers. https://books.google.com.au/books?hl=en&amp;lr=&amp;id=Yq_uDwAAQBAJ&amp;oi=fnd&amp;pg=PR7&amp;dq=How+can+betrayal+trauma+survivors+develop+trust+in+similar+situations+again%3F&amp;ots=qPVXjqErDK&amp;sig=MZhvi-7EBgNXJXGcfQrj4abtJ60&amp;redir_esc=y#v=onepage&amp;q&amp;f=false

Ward, R. (2026). Serial infidelity and psychological abuse: Diagnostic blind spots in recognising betrayal trauma. ''The International Journal of Psychiatry in Medicine''. [https://journals.sagepub.com/doi/10.1177/00912174261428855 https://doi.org/10.1177/00912174261428855]

Zurbriggen, E. L., &amp; Freyd, J. J. (2004). The link between childhood sexual abuse and risky sexual behavior: The role of dissociative tendencies, information-processing effects, and consensual sex decision mechanisms. In E. L. Zurbriggen &amp; J. J. Freyd, ''From child sexual abuse to adult sexual risk: Trauma, revictimization, and intervention'' (pp. pp. 135–157). https://pages.uoregon.edu/dynamic/jjf/articles/zf2004.pdf

}}

==External links==
[https://www.youtube.com/watch?v=5TUxH2izTGI Infidelity: to stay or go?] (Lucy Beresford, TEDX)

[https://www.youtube.com/watch?v=k8foEEWnsFw How to Rebuild Trust After Infidelity] (Samuel from Affair Recovery)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Trauma]]
[[Category:Motivation and emotion/Book/Trust]]</text>
      <sha1>gohpphoijonapsjxrm0jlt96tm3zs65</sha1>
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  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Dark empathy</title>
    <ns>1</ns>
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    <revision>
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      <timestamp>2026-11-10T07:45:19Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Book chapter feedback</comment>
      <origin>2845089</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16314" sha1="d0dpjpi04sqooyfrxd2685l81ya9iv9" xml:space="preserve">== Reference ==

Hi, This should be a pretty interesting chapter. I was going to add this link (not a journal article) directly to your external link section, but thought I'd leave it here just in case. Good luck with the page.  https://www.wikihow.com/Dark-Empath [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 08:31, 26 August 2026 (UTC)

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Heading structure --&gt;
&lt;!-- 2-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement
# Remove &quot;Case study&quot; heading (fixed)
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# Revise &quot;Dark empathy&quot; heading to be more descriptive
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# Present focus questions in a feature box at the end of this section
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&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Basic development
# The key points lack a clear definition of DE; there may be some conceptual confusion with the dark triad
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# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
&lt;!-- Theory and research --&gt;
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# Insufficient use of citations
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
# Consider using Studiosity Writing Feedback+ or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft.
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# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting]
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# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
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# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]]
# To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click &quot;compare selected revisions&quot;, and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:28, 14 September 2026 (UTC)

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# This is a basic chapter
# The main areas for potential improvement are:
## provide more detailed review of the best psychological research about the topic
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## more substantial engagement in social contribution
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# Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits
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# This chapter appears to be human-written
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# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
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# Basic use of tables, figures, and/or lists to convey key theoretical information
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## describing the methodology (e.g., sample, country, measures) in important studies
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## The chapter is well structured, with major sections using sub-sections
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## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour)
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      <text bytes="28318" sha1="jdl7wqndtsowcpwgzvnalihq9k172ky" xml:space="preserve">{{title|Self-blame and emotion:&lt;br&gt;How does self-blame influence emotional responses to negative events?}}

==Overview==
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[[File:Fort Wayne Daisies player, Marie Wegman, of the All American Girls Professional Baseball League arguing with umpire Norris Ward Opa-locka, Florida.jpg|thumb|right|250px| '''Figure 1'''. Two friends having an argument {{ic|Explain how this relates to self-blame and emotion}}]]

'''Imagine this...'''

You are having an argument with someone close to you (see Figure 1). Afterwards, you begin thinking about what went wrong and whether you could have handled the situation differently. You might think, &quot;I should have listened more carefully&quot; or &quot;I shouldn't have reacted that way.&quot; Alternatively, you might begin questioning yourself more personally, thinking, &quot;I'm a terrible friend&quot; or &quot;There's something wrong with me.&quot; Although both responses involve blaming yourself for the same negative experience, they focus on different aspects of responsibility.

These thoughts may also lead to different emotional responses. You might feel guilty about what you said and want to apologise, or you might feel ashamed and begin questioning your worth as a person. For some people, reflecting on the experience may encourage them to approach future disagreements differently. For others, repeatedly blaming themselves may make it difficult to move forward, particularly when they believe the problem is part of who they are.

This raises an important question: Why can blaming ourselves for a negative experience sometimes encourage positive change, while at other times contributing to ongoing emotional distress?  

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Self-blame is a common response to negative experiences, but its emotional consequences are not always straightforward. While some individuals may use these experiences to reflect on their actions and consider what they could do differently, others may develop negative beliefs about themselves that contribute to ongoing emotional distress. Psychological research suggests that the way individuals interpret their responsibility for an event can influence both their emotional responses and how they approach similar situations in the future.

Understanding this relationship is important because self-blame is not necessarily helpful or harmful in every situation. Its consequences may depend on how individuals explain the event, whether they believe they have control over future outcomes, and whether their response encourages constructive action or repeated negative thinking.

This chapter examines how self-blame influences emotional responses following negative events. It explores the different forms of self-blame, psychological theories that explain the relationship between self-blame and emotion, and research examining its emotional consequences. It also considers the circumstances in which self-blame may encourage positive change or contribute to ongoing distress. 

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'''Focus questions'''

* What are the different forms of self-blame? 
* How do psychological theories explain the relationship between self-blame and emotion? 
* How do different forms of self-blame influence emotional responses?
* Under what circumstances might self-blame be helpful or harmful?
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==What is self-blame?==
Self-blame occurs when someone believes they are responsible for a negative event or outcome. When something goes wrong, people often try to understand why it happened and whether they could have prevented it. Sometimes, this involves thinking about their own actions, while other times, they may question their personal abilities or characteristics. Although both involve blaming oneself, the difference is important because it may influence how much control someone believes they have over similar situations in the future. {{f}}

== What are the different forms of self-blame? ==
Janoff-Bulman (1979) distinguishes between the [https://psycnet.apa.org/doiLanding?doi=10.1037%2F0022-3514.37.10.1798 two forms of self-blame], behavioural self-blame and characterological self-blame. These differ based on whether a person blames something they did or something they believe is wrong with them as a person.

=== Behavioural self-blame:===
Behavioural self-blame occurs when someone believes a negative event happened because of something they did or failed to do which directly impacted the outcome of the event. For example, an employee who misses an important work deadline may blame themselves for not managing their time effectively. Rather than believing they are incapable of doing their job, they recognise that their preparation or time management may have contributed to the outcome. 
Janoff-Bulman (1979) suggested that behavioural self-blame may help individuals maintain a sense of control because their actions can potentially be changed. If someone believes they could have prepared differently, they may also believe they can improve their approach next time. This does not mean that behavioural self-blame is always helpful, as someone may still feel guilty or continue blaming themselves for a mistake. Research by Anderson et al. (1994) challenges the assumption that behavioural self-blame is necessarily beneficial (see Figure 2).

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Did you know {{g}}💡

Anderson et al. (1994) examined {{how}} two samples of university students {{where}}, involving 1,001 participants in total. Their findings indicated that both behavioural and characterological self-blame were {{ic|how strongly?}} associated with depression and loneliness. This suggests that focusing on one's actions rather than personal characteristics does not necessarily protect someone from emotional difficulties.
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=== Characterological self-blame: ===
Characterological self-blame occurs when someone believes a negative event happened because of their personal characteristics or abilities. Rather than focusing on a particular action, they may interpret the experience as evidence of something negative about themselves. For example, an employee who receives negative feedback from their manager may believe they are incompetent or simply not good enough at their job. Rather than considering whether there are specific areas they could improve, they interpret the feedback as evidence that they lack the personal abilities needed to succeed.
Janoff-Bulman (1979) implied that characterological self-blame can be particularly distressing because people may view these characteristics as difficult to change. If someone believes the problem is part of who they are, they may feel that similar negative outcomes are likely to happen again. 

This distinction helps explain why two people can blame themselves for a similar experience but respond differently. However, behavioural self-blame should not automatically be considered positive or characterological self-blame the only harmful form. {{ic|What type of?}} Research by Anderson et al. (1994) suggests that both forms can be associated with emotional difficulties, meaning the consequences of self-blame may depend on more than what a person believes caused the event.

{{ic|Consider putting this quiz question in a feature box}}

Choose your answers and click &quot;Submit&quot;: 
&lt;quiz display=simple&gt;
{Characterological self-blame focuses on specific behaviours that can be changed to prevent similar negative outcomes in the future:
|type=&quot;()&quot;}
- True
+ False
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== How do psychological theories explain the relationship between self-blame and emotion? ==
People do not always respond emotionally to negative experiences in the same way. Although two people may experience a similar situation, their emotions can differ depending on how they understand what happened. For example, two employees who make a mistake at work may both blame themselves, but one might feel guilty about their actions while the other feels ashamed of their abilities. This suggests that emotional responses are influenced not only by the negative event itself, but also by how a person explains and interprets it. 

Two psychological approaches that help explain this relationship are attribution theory and cognitive appraisal theory. Whilst they focus on different aspects of emotional responses, both suggest that the meaning a person gives to a negative experience is important. 

=== Attribution theory ===
[https://psycnet.apa.org/doiLanding?doi=10.1037%2F0033-295X.92.4.548 Attribution theory] focuses on how people explain why events happen. Weiner (1985) proposed that these explanations can be understood through three dimensions, locus, stability and controllability. 

Locus refers to whether someone believes the cause of an event comes from themselves or from an external factor. Stability refers to whether the cause is seen as something temporary or likely to remain the same. Controllability considers whether the person believes they can influence the cause of the event. 
These dimensions help explain why self-blame can lead to different emotional responses. For example, someone who believes they missed an important deadline because they did not manage their time effectively may feel guilty but also believe they can prevent the same mistake in the future. However, someone who believes they missed the deadline because they are naturally incompetent may feel ashamed and less confident about improving. 

Both individuals blame themselves, but they understand the cause of the mistake differently. This is important because Weiner's (1985) theory suggests that emotional responses depend partly on whether the perceived cause is stable and controllable, rather than simply whether a person believes they are responsible.

{{ic|Is there any research about attribution theory and self-blame?}}

=== Cognitive appraisal theory === 
Cognitive appraisal theory {{f}} focuses on how people evaluate the meaning of an experience. Rather than suggesting that an event automatically produces a particular emotion, this approach considers how an individual interprets the situation and what it means for them. 

For example, receiving negative feedback on a sporting performance by a coach may be viewed as an opportunity to improve or as evidence of personal failure. Someone who believes the feedback identifies an area they can improve may feel disappointed but remain motivated. In comparison, someone who interprets the feedback as evidence they are not capable of succeeding may experience shame or anxiety about making further mistakes. 

Tracy and Robins (2006) examined how different interpretations of negative experiences relate to guilt and shame {{ic|sample? culture? methods?}}. Their findings supported the idea that shame is more closely associated with explanations involving behaviours that can potentially be changed. This research helps explain why self-blame {{ic|how do guilt and shame relate to self-blame?}} does not always produce the same emotional response. The way people understand their responsibility can influence whether they focus on correcting a particular mistake or develop more negative feelings about themselves. 

Overall, attribution theory and cognitive appraisal theory help explain why self-blame can influence emotions differently. Attribution theory examines what people believe caused an event, while cognitive appraisal theory considers how they evaluate the meaning of the event. Together, these approaches suggest that understanding emotional responses to self-blame requires considering both the explanation for what happened and what that explanation means to the individual.

== How do different forms of self-blame influence emotional responses? == 
Self-blame can be associated with several emotional responses, including guilt, shame, sadness and anxiety. However, these emotions are not necessarily experienced in the same way. The distinction between behavioural and characterological self-blame is important because it helps explain whether someone focuses on a particular mistake or develops negative beliefs about themselves. Research suggests that these different interpretations can influence both emotions and peoples experience and how they respond to negative events {{f}}. 

=== Guilt and shame ===
Guilt and shame are both emotions that can arise when someone believes they have done something wrong. However, guilt generally involves a negative evaluation of a particular action, while shame involves a negative evaluation of oneself.

For example, someone who forgets an important friend's birthday might feel guilty about being forgetful and decide to make it up to them. However, if they interpret the mistake as evidence that they are a terrible friend, they may experience shame. Although both reactions involve self-blame, the second involves a broader judgement about the person's character. Tracy and Robins (2006) found support {{how}} for the relationship between different types of appraisals and experiences of guilt and shame. Their research suggests that shame is more closely associated with negative evaluations of the self, whereas guilt is more closely associated with evaluations of particular behaviours.

Tilghman-Osborne et al. (2008) further examined this relationship in a [https://guilfordjournals.com/doi/10.1521/jscp.2008.27.8.809 longitudinal study] involving 221 adolescents {{where}}. The researchers found that shame and characterological self-blame were {{how}} closely related, suggesting that they may reflect a similar underlying pattern of negative self-evaluation. However, the relationship between guilt and behavioural self-blame was less consistent. This is important because it suggests that the distinction between the two forms of self-blame does not perfectly explain which emotion someone will experience. Although behavioural self-blame and guilt share similarities, they should not automatically be treated as the same experience.

{{ic|Is there any research about attribution theory and self-blame?}}

Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Research suggests that characterological self-blame is closely associated with feelings of shame: 
|type=&quot;()&quot;}
+ True
- False
&lt;/quiz&gt;

=== Sadness, anxiety, and depressive symptoms === 
Beyond guilt and shame, self-blame may also contribute to broader emotional difficulties. When someone repeatedly interprets negative experiences as evidence of personal inadequacy, these beliefs may affect how they feel about themselves and their expectations for the future.

For example, a student who receives several disappointing grades might begin questioning their ability to succeed at university. Rather than feeling disappointed about one assessment, they may become increasingly discouraged and anxious about future results. This illustrates how negative self-evaluations can extend beyond an immediate emotional reaction. Janoff-Bulman (1979) found {{how}} that depressed female university students {{where}} reported greater characterological self-blame than non-depressed students, while behavioural self-blame did not differ significantly between the {{which}} groups. This suggested that blaming negative experiences on personal characteristics may be particularly relevant to depressive symptoms.

However, subsequent research has complicated this distinction. Anderson et al. (1994) found {{how}} associations between both forms of self-blame and emotional difficulties, including depression and loneliness. Similarly, Tilghman-Osborne et al. (2008) identified {{how}} a significant {{ic|how strong?}} relationship between shame, characterological self-blame and depressive symptoms among adolescents. Together, these studies suggest that negative judgements about the self may be particularly important in understanding depressive symptoms. However, they also show that behavioural self-blame should not automatically be considered harmless. The relationship may depend on how people interpret their responsibility and whether self-blame becomes a repeated pattern. There are also limitations to consider. Much of this research examines associations between self-blame and depressive symptoms rather than establishing a direct cause-and-effect relationship. The studies also involve different populations, including university students and adolescents, which may limit how broadly their findings can be applied.

Further evidence comes from [https://psycnet.apa.org/doiLanding?doi=10.1037%2Fa0021466 Kim et al. (2011)], who conducted a meta-analysis of 108 studies involving 22,411 participants. They found that shame had a stronger association with depressive symptoms (r = .43) than guilt (r = .28). However, guilt involving exaggerated responsibility for events beyond a person's control was also strongly associated with depressive symptoms (r = .39). These findings suggest that emotional difficulties may depend not only on whether someone experiences guilt or shame, but also on how they interpret their responsibility for a negative event.

Overall, research indicates that different forms of self-blame are related to emotional responses, particularly guilt, shame and depressive symptoms. However, the findings suggest that these relationships are more complex than simply associating behavioural self-blame with guilt and characterological self-blame with shame.

== Under what circumstances might self-blame be helpful or harmful? ==
The consequences of self-blame may depend on what happens after someone accepts responsibility for a negative experience. In some situations, recognising a mistake can encourage reflection and improvement. In others, blaming oneself may make it more difficult to move forward, especially when the person continues focusing on what went wrong. Understanding the difference is important because recognising responsibility does not necessarily lead to a helpful emotional response.

=== When self-blame encourages constructive change ===
Self-blame may be helpful when someone uses the experience to identify what they could do differently in the future. For example, a person who performs poorly in a job interview might recognise that they did not prepare enough. Although they may feel disappointed, they can use this experience to practise their interview skills and approach future opportunities differently.

{{ic|What type of?}} Research by Tangney et al. (2007) suggests that guilt can encourage constructive responses following negative experiences. Their review discusses how guilt is often associated with taking responsibility and attempting to repair mistakes, such as apologising after hurting someone. This suggests that recognising responsibility may encourage positive change when attention is directed towards what can be done differently in the future. However, guilt is not necessarily helpful when someone takes responsibility for circumstances beyond their control. An important factor is whether the person believes they have control over what happens next. Weiner's (1985) attribution theory suggests that perceived controllability influences how people respond to negative outcomes. If someone believes their actions can make a difference, recognising a mistake may encourage them to take steps towards improving the situation.

However, there is a difference between accepting responsibility and unnecessarily blaming oneself. Recognising that a particular decision contributed to an outcome can be useful, but this does not mean that a person needs to continue feeling guilty to make a positive change. What may be helpful is the opportunity to learn from the experience, rather than the negative emotion itself. {{f}}

=== When self-blame contributes to ongoing distress === 
Self-blame may become harmful when someone repeatedly thinks about a negative experience without reaching a useful conclusion. Instead of identifying a way forward, they may continue questioning what they should have done differently. This pattern resembles rumination, which involves repeatedly focusing on distressing experiences and their possible causes or consequences. Nolen-Hoeksema et al. (2008) reviewed research showing that [https://journals.sagepub.com/doi/10.1111/j.1745-6924.2008.00088.xrumination rumination] is {{ic|how strongly?}} associated with prolonged negative mood and greater vulnerability to depressive symptoms. Although rumination and self-blame are not the same process, the findings help explain why repeatedly revisiting perceived personal mistakes may contribute to ongoing emotional difficulties.

The potential harm of self-blame is not limited to repeatedly thinking about negative experiences. Frazier (1990) found {{how}} that both behavioural and characterological self-blame were associated with {{ic|how much?}} greater depressive symptoms among {{ic|how many?}} women {{where}} who had experienced sexual assault. These findings challenge the assumption that focusing on one's behaviour is necessarily less harmful than blaming personal characteristics. However, because the study examined responses to a traumatic experience, its findings may not apply in the same way to everyday mistakes or disappointing performance. Another concern is that people do not always accurately judge how much responsibility they had for an event. For example, someone whose friendship ends may blame themselves entirely, even though the relationship was influenced by both people. If they overlook circumstances outside their control, their self-blame may reinforce feelings of guilt or inadequacy without helping them understand what happened. This highlights an important limitation in treating self-blame as a way of improving future outcomes. Even when someone believes they could have prevented an event, that belief does not necessarily mean they had the ability to do so. Self-blame may therefore be particularly unhelpful when it involves unrealistic responsibility or repeated negative thoughts that do not lead to constructive action.

Overall, the distinction between helpful and harmful self-blame is not simply whether someone accepts responsibility. It also depends on whether that responsibility is realistic, whether the person has control over future outcomes, and whether reflecting on the experience encourages constructive action. Recognising a mistake may provide an opportunity for change, whereas repeatedly blaming oneself for circumstances beyond one's control may prolong emotional distress.

==Conclusion==
Self-blame is more complex than simply believing that a negative event was one's own fault. How individuals interpret their responsibility can influence the emotions they experience, particularly when they attribute an outcome to their behaviour or personal characteristics. The distinction between behavioural and characterological self-blame helps explain why people may respond differently to similar experiences. Attribution theory and cognitive appraisal theory further demonstrate that emotional responses are influenced not only by what happens, but also by how individuals understand the cause and meaning of an event.

Research highlights the relationship between self-blame and emotions such as guilt, shame and depressive symptoms. While behavioural self-blame is often considered more constructive because it focuses on actions that may be changed, the evidence suggests that this distinction is not always straightforward. Both forms of self-blame have been associated with emotional difficulties, challenging the assumption that focusing on one's behaviour necessarily protects against distress. Similarly, although guilt may encourage individuals to take responsibility and repair mistakes, it can also become distressing when someone assumes responsibility for circumstances beyond their control.

Ultimately, whether self-blame is helpful or harmful may depend on how realistically individuals assess their responsibility and what they do with that understanding. Recognising a mistake may encourage reflection and positive change, particularly when future outcomes can be influenced by different actions. However, repeated negative thoughts or blaming oneself for uncontrollable circumstances may contribute to ongoing emotional distress. Understanding these differences is important because it shows that taking responsibility does not have to involve continued self-criticism. Instead, learning from negative experiences may be more constructive when individuals recognise what they can change without assuming responsibility for everything that went wrong.

==See also==
{{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

[https://en.wikipedia.org/wiki/Self-blame_(psychology) Self-blame]

[https://en.wikipedia.org/wiki/Attribution_(psychology) Attribution theory]

[https://en.wikipedia.org/wiki/Cognitive_appraisal Cognitive appraisal theory]

==References==
{{Hanging indent|1=

Abela, J. R. Z., Webb, C. A., Wagner, C., Ho, M.-H. R., &amp; Adams, P. (2006)'' The role of self-criticism, dependency, and hassles in the course of depressive illness: A multiwave longitudinal study. Personality and Social Psychology Bulletin, 32(3), 328–338.'' https://doi.org/10.1177/0146167205280911

Anderson, C. A., et al. ''“Behavioral and Characterological Attributional Styles as Predictors of Depression and Loneliness:'' Review, Refinement, and Test.” Journal of Personality and Social Psychology, vol. 66, no. 3, 1 Mar. 1994, pp. 549–558, https://pubmed.ncbi.nlm.nih.gov/8169764/

Frazier, P. A. (1990). ''Victim attributions and post-rape trauma. Journal of Personality and Social Psychology, 59(2), 298–304.'' https://doi.org/10.1037/0022-3514.59.2.298

Janoff-Bulman, R. (1979).''Characterological versus behavioral self-blame: Inquiries into depression and rape.'' Journal of Personality and Social Psychology, 37(10), 1798–1809. https://doi.org/10.1037/0022-3514.37.10.1798

''Kim, S., Thibodeau, R., &amp; Jorgensen, R. S. (2011). Shame, guilt, and depressive symptoms: A meta-analytic review. Psychological Bulletin, 137(1), 68–96.'' https://doi.org/10.1037/a0021466

''Lazarus, R. S. (1991). Emotion and adaptation. Oxford University Press.''

National Study of Mental Health and Wellbeing, 2020-21. (2022, July 22). Australian Bureau of Statistics. https://www.abs.gov.au/statistics/health/mental-health/national-study-mental-health-and-wellbeing/2020-21?utm_ 

Nolen-Hoeksema, S., Wisco, B. E., &amp; Lyubomirsky, S. (2008). ''Rethinking rumination. Perspectives on Psychological Science, 3(5), 400–424.'' https://doi.org/10.1111/j.1745-6924.2008.00088.x

Peterson, C., Schwartz, S. M., &amp; Seligman, M. E. P. (1981). ''Self-blame and depressive symptoms. Journal of Personality and Social Psychology,'' 41(2), 253–259. https://doi.org/10.1037/0022-3514.41.2.253

Tangney, J. P., Stuewig, J., &amp; Mashek, D. J. (2007).''Moral emotions and moral behavior. Annual Review of Psychology, 58, 345–372.'' https://doi.org/10.1146/annurev.psych.56.091103.070145

Tilghman-Osborne, C., Cole, D. A., Felton, J. W., &amp; Ciesla, J. A. (2008). ''Relation of guilt, shame, behavioral and characterological self-blame to depressive symptoms in adolescents over time. Journal of Social and Clinical Psychology'', 27(8), 809–842. https://doi.org/10.1521/jscp.2008.27.8.809

Tracy, J. L., &amp; Robins, R. W. (2006).''Appraisal antecedents of shame and guilt: Support for a theoretical model. Personality and Social Psychology Bulletin, 32(10), 1339–1351.'' https://doi.org/10.1177/0146167206290212

Weiner, B. (1985).''An attributional theory of achievement motivation and emotion. Psychological Review, 92(4), 548–573.'' https://doi.org/10.1037/0033-295X.92.4.548

}}

==External links==
* [https://www.cci.health.wa.gov.au/~/media/CCI/Consumer-Modules/Back-from-The-Bluez/Back-from-the-Bluez---05---Unhelpful-Thinking-Styles.pdf Unhelpful thinking styles] (Government of Western Australia) 
* [https://www.psychologytools.com/tools/digital-tools/self-blame?language=en-us Self-blame] (Psychology tools) 
* [https://www.mind.org.uk/information-support/types-of-mental-health-problems/trauma/effects-of-trauma/ Understanding the effects of trauma] (Mind) 

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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      <text bytes="17358" sha1="94a0o2bp3acbpql4cwo6d29tixdls3r" xml:space="preserve">== General comments ==

* The opening scenario is engaging and provides a clear introduction to how moral disgust may influence juror decision-making.
* The chapter has a logical structure and the headings cover important areas such as moral disgust, juror judgements, moderating factors, and practical implications.
* Some sections would benefit from further development and stronger integration of psychological theory and empirical research.
* There are still some unfinished areas and placeholders, such as the proposed evidence table, which should be completed before the final submission.
* The conclusion should clearly answer the chapter question and summarise the main take-home messages from the research.
* Overall, the chapter has a strong foundation and an interesting topic, but further evidence, citations, and refinement would strengthen the final version.

[[User:Karabi Tasneem|Karabi Tasneem]] ([[User talk:Karabi Tasneem|discuss]] • [[Special:Contributions/Karabi Tasneem|contribs]]) 21:48, 27 August 2026 (UTC)--[[User:Karabi Tasneem|Karabi Tasneem]] ([[User talk:Karabi Tasneem|discuss]] • [[Special:Contributions/Karabi Tasneem|contribs]]) 21:50, 27 August 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:Yellowvines|Hi [[User:Yellowvines|Yellowvines]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:34, 10 September 2026 (UTC)
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:39, 10 November 2026 (UTC)</text>
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      <text bytes="22774" sha1="7gfrw3edv93c0xe24qbloohejwnisl0" xml:space="preserve">{{title|Relatedness motivation in self-determination theory:&lt;br&gt;How does the need for relatedness function within self-determination theory to shape motivation and behaviour?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File:Raise hand gesture occupy.svg|Raise_hand_gesture_occupy|right|140px|thumb|'''Figure 1'''. A raised hand: Lauren's tutor encouraged students to ask for help and connect with each other.]]

;Imagine this...

Lauren is a first-year university student studying psychology who recently started a demanding course. During the first few weeks, she finds the workload difficult and is unsure whether she is capable of succeeding. She chose to study psychology because she is interested in the subject, however, she feels isolated because her classmates seem to already know one another. 

One of Lauren’s tutors makes an effort to learn about students, encourages them to contribute to discussions, and provides feedback. The tutor also organises small-group activities where students can get to know one another and emphasises that asking for help is a normal part of learning. Lauren engages with these opportunities. Over time, Lauren begins to feel that she belongs in the course and that both her tutor and classmates value her participation.

As Lauren feels increasingly connected to the people around her, she becomes more willing to attend tutorials, participate in discussions, ask questions (see Figure 1), and seek feedback when she does not understand something. She also starts studying with classmates outside class. Lauren is no longer completing the course requirements only because she feels she has to. Her growing sense of belonging helps her internalise the value of the course, so she begins to think, “Psychology is important to me, and I want to do well because I see myself as part of this community.” This shift leads to more persistent, engaged behaviour.

{{RoundBoxBottom}}

Self-determination theory (SDT) is a psychological theory that explains how social and environmental factors influence human motivation, behaviour, personal development, and psychological well-being (Ryan &amp; Deci, 1985). According to SDT, three basic psychological needs—autonomy, competence, and relatedness—{{g}}are important for psychological growth and optimal functioning (Ryan &amp; Deci, 2000). Autonomy refers to experiencing a sense of choice and personal endorsement over one’s actions, while competence involves feeling capable and effective when interacting with the environment. Relatedness refers to feeling connected to others and experiencing a sense of belonging, care, and mutual respect (Ryan &amp; Deci, 2000). Although all three needs contribute to motivation, this chapter focuses on how the satisfaction and frustration of relatedness influence the development of motivation and the behaviours that follow.

Within SDT, relatedness can influence motivation by shaping how individuals experience and internalise the activities, expectations, and values present in their social environments (Ryan &amp; Deci, 2000; Slemp et. al, 2024). When individuals feel accepted, supported, and valued by others, they may be more willing to participate in activities, identify with their purpose, and sustain their involvement. Autonomous motivation can include both intrinsic motivation, in which an activity is undertaken because it is enjoyable or interesting, and internalised extrinsic motivation, in which an activity is undertaken because its value has been personally accepted (Ryan &amp; Deci, 2000). Research has linked relatedness satisfaction with students’ identified regulation {{explain}} and subsequent behavioural engagement in mathematics (Hofverberg et al., 2022). Relatedness support has also been associated with improvements in adolescents’ motivation, positive affect, and motor learning (Kaefer &amp; Chiviacowsky, 2021).

In contrast, relatedness frustration may occur when individuals experience exclusion, rejection, or a lack of meaningful social connection. Such experiences can undermine motivation and well-being and may contribute to reduced engagement or withdrawal (Hofverberg et al., 2022; Slemp et. al., 2024). However, the effects of relatedness are not uniform across individuals or contexts, and relatedness should not be considered independently of autonomy and competence. Research on interpersonal support for basic psychological needs indicates that supportive social environments are associated with greater need satisfaction, well-being, and performance (Slemp et al., 2024). Understanding relatedness therefore requires consideration of the relationships and environments in which individuals learn, work, and interact with others.

This chapter examines the role of relatedness within SDT and considers how relatedness satisfaction and frustration shape autonomous motivation and behaviour. It explores relevant theoretical concepts and mini-theories of SDT, including basic psychological needs theory and relationship motivation theory, before examining research findings across contexts. The chapter also considers how relationships and social environments can support or thwart relatedness, and discusses the implications for engagement, persistence, well-being, and effective participation. Overall, the chapter aims to explain why meaningful social connections are an important part of understanding human motivation and behaviour.

{{RoundBoxTop|theme=3}}
'''Focus questions'''

* What is the need for relatedness within Self-Determination Theory, and why is it important in shaping motivation?
* How does relatedness satisfaction influence the development of autonomous motivation?
* How does relatedness satisfaction or frustration affect individuals’ behaviour, such as engagement, persistence, and disengagement?
* How do relationships and social environments affect relatedness, and what are the implications on motivation and behaviour?
{{RoundBoxBottom}}

==Self-Determination Theory==
[[w:Self determination theory|Self-determination theory]] is a macro-theory of psychology created by [[w:Edward Deci|Edward Deci]] and [[w:Richard Ryan|Richard Ryan]] relating to motivation, behaviour, and personality. Deci and Ryan published their first comprehensive work about SDT in 1985. SDT utilises an organismic-dialectical approach, which assumes people are ‘active organisms’ who have tendencies toward growth, overcoming challenges, and integrating new experiences into their sense of self. SDT posits this inherent tendency toward growth as well as peoples’ innate psychological needs are the basis for self-motivation and personality integration. Ryan and Deci (2000), {{g}} propose that humans have three basic psychological needs which include autonomy, competence, and relatedness and when these needs are met, they are essential to optimal personal growth, wellbeing, and [[W:intrinsic motivation|intrinsic motivation]]. SDT also considers the role of social or cultural factors in either supporting or 'thwarting' individuals' motivation, well-being, and personal growth. {{g}} (Ryan &amp; Deci, 2000).  

=== Basic psychological needs theory ===
Basic psychological needs theory is a micro-theory within SDT. BPNT refers to the idea in SDT that people have three basic psychological needs that are central to wellbeing, growth and intrinsic motivation (Ryan &amp; Deci, 2000). [[w:Autonomy|Autonomy]] is the experience of ones actions as their own and personally-endorsed, as opposed to being controlled or imposed upon them. [[w:Competence|Competence]] is closely related to [[w:Self-Efficacy|self-efficacy]] and refers to the need to feel capable and effective and confident in your interactions with your environment, and to master challenges and achieve goals. 

Relatedness is the innate human need to develop meaningful relationships with those around you, and to feel a sense of mutual care, respect, and warmth between yourself and others. Relatedness also refers to experiencing a sense of belonging, community, and kinship with others (Ryan &amp; Deci, 2000).

=== Relationship motivation theory ===
Relationship motivation theory (RMT) is the sixth micro or mini theory within SDT that explains the development, function, and success of relationships (Deci &amp; Ryan, 2000). RMT demonstrates that relatedness satisfaction alone is not sufficient for a successful relationship. In relationships, the need for relatedness, autonomy, and competence must all be met, and both partners must support each others needs. Deci &amp; Ryan (2000) note that although the need for relatedness is experiencing connection and belonging, a meaningful relationship is tied to the satisfaction of all three needs, and mutual-support of each others' needs. Knee et. al {{g}} (2014) note that SDT can be viewed as the framework for an optimal romantic relationship. Romantic partners with relationship-contingent self-esteem leads to emotional insecurity, whereas partners with non-contingent self-esteem experience greater intimacy (Knee et. al., 2014) {{g}}.

==Cognitive evaluation theory==
[[File:Muhammad Intrinsic vs Extrinsic Motivation.png|200px|thumb|'''Figure 2'''. The player on the left wants to win the trophy which is an example of extrinsic motivation. The player on the right is playing for his own enjoyment, indicating he is intrinsically motivated. {{ic|Explain how this relates to CET.}}]]

Cognitive evaluation theory (CET) is a micro-theory within self-determination theory. Cognitive evaluation theory is essential in understanding motivation within SDT, specifically, intrinsic motivation which refers to motivation to engage in a task out of an inherent interest, enjoyment, and engagement, whereas extrinsic motivation refers to an individual being motivated by external rewards or reinforcement (Ryan &amp; Deci, 2000). CET suggests that intrinsic motivation is more valuable than extrinsic motivation, as those that are intrinsically motivated feel a sense of choice over what they are doing and gain a feeling of competence and enjoyment from the activities or tasks they complete. {{g}} (Hagger et. al., 2020). The role of social and environmental influences such as rewards, feedback and communications, can either facilitate or undermine intrinsic motivation. 

CET considers intrinsic motivation to be socially and environmentally influenced. For example, when an individual receives positive feedback, they are more likely to be intrinsically motivated, and when an individual receives negative feedback, they are less likely to be intrinsically motivated (Ryan &amp; Deci, 2000). CET also aims to specify that factors that explain the variability of intrinsic motivation. 

=== Taxonomy of human motivation ===
Deci &amp; Ryan (2000) propose a taxonomy of motivation (shown in the figure below {{ic|use APA style}}) to explain for the continuum between extrinsic and intrinsic motivation.

[[File:Taxonomy of motivation.png|center|600x600px|thumb|'''Figure 3.'''Ryan &amp; Deci's (2000) taxonomy of moitvation. A visual spectrum of motivation]]

The taxonomy of motivation highlights motivation as sitting on a spectrum of autonomy, beginning with a complete lack of motivation (motivation) and ending with doing an activity purely for its inherent satisfaction or interest (intrinsic regulation). This model also highlights the differences between extrinsic and intrinsic motivation versus controlled and autonomous motivation, introducing regulations to specify the nuances of motivation. Notably, autonomous motivation spans across both extrinsic and intrinsic motivation (Deci &amp; Ryan, 2000).

'''Quiz'''

Choose your answers and click &quot;Submit&quot;:

&lt;quiz display=&quot;simple&quot;&gt;
{Autonomous motivation is the same as intrinsic motivation:
|type=&quot;(false)&quot;}
- True
+ False

{Winning a trophy is an example of extrinsic motivation, and participating in a hobby that interests themselves is an example of intrinsic motivation:
|type=&quot;(true)&quot;}
+ True
- False
&lt;/quiz&gt;

==Relatedness, motivation, and behaviour==
CET focuses on the fundamental psychological needs for autonomy and competence and how they can enhance or thwart intrinsic motivation (Ryan &amp; Deci, 2000). SDT specifies that needs-supportive environments should lead to self-determined student motivation (Burea et. al., 2021). In a meta-analysis, Bureau et. al. (2021) found that competence was the strongest predictor for self-determined motivation, followed by autonomy and then relatedness. Although CET focuses on autonomy and competence as the main catalysts for intrinsic motivation, Ryan &amp; Powelson (1991) explain that relatedness needs are not antithetical to competence or autonomy and suggests people feel most strongly related to people who are responsive to your expressions of autonomy. Relatedness plays a role especially in the social and environmental contexts which influence motivation and behaviour (Hagger et. al., 2020). The effects of relatedness satisfaction and frustration on motivation and behaviour has been investigated in literature thoroughly {{vague}} {{expand}}.

=== Wellbeing ===
The role of wellbeing in the relationship between relatedness, motivation and behaviour has been investigated. Slemp et. al {{g}} (2024), found {{how}} that having interpersonal supports for autonomy, competence, and relatedness were associated with the satisfaction of those psychological needs. Environmental support for psychological needs were associated {{ic|how strongly?}} with greater wellbeing, which then lead to improved performance (Slemp et. al., 2024).

To test the effects of relatedness satisfaction and relatedness frustration on task learning, Kaefer &amp; Chiviacowsky (2021) conducted an experiment in which {{ic|how many?}} adolescents {{where}} were either put in a relatedness support condition, relatedness frustration condition, or a control group and tasks {{g}} with learning how to hit a ball with a racket toward a target. Results showed that relatedness supports not only enhanced {{ic|how strongly?}} the task-learning of adolescents and affected motivation, but influence the affect of participants, suggesting that wellbeing was a mediating factor in task-learning {{g}}.

=== Education ===
[[File:A public high school teacher in a classroom in the United States 13.jpg|right|330px|thumb|'''Figure 3'''. Teacher relatedness can increase participation in education (Collie, 2024).]]

Relatedness has been found {{how}} to influence {{ic|how strongly?}} students' engagement and performance in educational settings (Kaefer &amp; Chiviacowsky, 2021; Hofverberg et. al., 2022). Behavioural engagement in education is essential to students' learning (Hofverberg et. al. {{g}}, 2022). Hofverberg et. al. {{g}} (2022) found that perceived relatedness satisfaction of students was positively {{ic|how strongly?}} associated with increased behavioural engagement in mathematics. When relatedness is supported, task-learning is perceived by students as more socially and emotionally meaningful and therefore motivates participation behaviours (Kaefer &amp; Chiviacowsky, 2021). Additionally, students; {{g}} perceptions of relatedness with their teachers has been associated with better homework practices and lower behavioural disaffection (Collie, 2024).

Burea et. al. (2021), {{g}} note that the weaker role of relatedness in self-determined motivation in education may be context-dependant, noting that relatedness may more strongly influence self-determined motivation in leisure contexts. Hofverberg et. al.  {{g}} (2022) notes that although the literature commonly investigate {{g}} links between the satisfaction of basic psychological needs and students' motivational regulations, but literature that utilises the full-model of SDT is limited. Specifically, more detailed research into the full chain of events that leads to higher quality engagement in education {{g}}. Hofverberg et. al. (2022) found {{how}} that relatedness satisfaction predicted identified regulation which then, in turn, predicted engagement in mathematics. These findings highlight the importance of investigating the role of regulation in the understanding how relatedness functions within SDT to shape motivation and behaviour.

==Conclusion==
Relatedness is a key psychological need within self-determination theory, and it influences individuals' motivation and engagement with their environment. When individuals experience meaningful relationships and a sense of belonging and mutual care, they are more likely to be autonomously motivated (Hovferberg. et. al., 2022). Hovferberg found that relatedness satisfaction predicted students' identified regulation {{explain}}, which subsequently predicted {{ic|how strongly?}} behavioural engagement in mathematics. Although relatedness contributes to shaping motivation, the satisfaction of competence and autonomy in conjunction with relatedness satisfaction are essential to self-determined motivation (Ryan &amp; Deci, 2000). 

Relatedness satisfaction has been found to influence behaviours such as engagement and persistence (Slemp et. al., 2024). Individuals are more likely to experience positive emotions when relatedness is satisfied as well as being more likely to participate and persist with challenging tasks (Slemp et. al., 2024). Relatedness support was found to enhance adolescents' motivation, positive affect, and motor learning. When the adolescents experienced a lack or belonging and support from peers, an decrease in participation and motivation may be undermined (Kaefer &amp; Chiviacowsky, 2021). Relatedness frustration can therefore contribute to disengagement, although behavioural outcomes also depend on other psychological needs being satisfied and contextual factors (Burea et. al., 2021).

Relationships and social environments play a significant role in determining whether relatedness is satisfied or frustrated (Ryan &amp; Deci, 2000). Supportive relationships with teachers, peers, family members, colleagues, and romantic partners can foster belonging and create environments in which individuals feel valued and encouraged (Ryan &amp; Deci, 2000). Collie (2024) associated students’ social-emotional need satisfaction with academic engagement and social-emotional skills, while Slemp et al. (2024) found that interpersonal support for basic psychological needs was associated with greater need satisfaction and well-being, as well as improved performance. These findings highlight the practical importance of creating inclusive, respectful, and supportive social environments in educational, workplace, and interpersonal settings.

Relatedness not only shapes the extent of motivation but also its quality and its expression in behaviour (Burea et. al, {{g}} 2021; Slemp et. al, 2024). Fostering meaningful connections can support autonomous motivation, well-being, engagement, and persistence, whereas environments that undermine belonging may weaken these outcomes (Collie, 2024; Kaefer &amp; Chiviacowsky, 2021). However, relatedness should not be considered in isolation. Supporting autonomy and competence alongside relatedness is important for promoting sustained motivation, positive development, and effective functioning (Ryan &amp; Deci, 2000; Ryan &amp; Powelson, 1991).

==See also==
*[[w:Cognitive evaluation theory|Cognitive evaluation theory]] (Wikipedia)
*[[w:Extrinsic motivation and psychological needs|Extrinsic motivation and psychological needs]] (Wikipedia)
*[[w:Intrinsic motivation|Intrinsic motivation]] (Wikipedia)
*[[w:Social relation|Relatedness]] (Wikipedia) 
*[[w:Self determination theory|Self-determination theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Bureau, J. S., Howard, J. L., Chong, J. X. Y., &amp; Guay, F. (2022). Pathways to Student Motivation: A Meta-Analysis of Antecedents of Autonomous and Controlled Motivations. Review of Educational Research, 92(1), 46-72. https://doi.org/10.3102/00346543211042426

Collie, R. J. (2024). Social-emotional need satisfaction and students’ academic engagement and social-emotional skills. Educational Psychology, 44(2), 117–135. https://doi.org/10.1080/01443410.2024.2324751

Hagger, M. S., Hankonen, N., Chatzisarantis, N. L. D., &amp; Ryan, R. M. (2020). Changing Behavior Using Self-Determination Theory. In M. S. Hagger, L. D. Cameron, K. Hamilton, N. Hankonen, &amp; T. Lintunen (Eds.), The Handbook of Behaviour Change (pp. 104–119). Cambridge University Press. https://doi.org/10.1017/9781108677318.008

Hofverberg, A., Winberg, M., Palmberg, B., Andersson, C., &amp; Palm, T. (2022). Relationships between basic psychological need satisfaction, regulations, and behavioral engagement in mathematics. Frontiers in Psychology, 13. https://doi.org/10.3389/fpsyg.2022.829958 

Kaefer, A., &amp; Chiviacowsky, S. (2021). Relatedness support enhances motivation, positive affect, and motor learning in adolescents. Human Movement Science, 79, 102864. https://doi.org/10.1016/j.humov.2021.102864

Knee, C. R., Hadden, B. W., Porter, B., &amp; Rodriguez, L. M. (2013). Self-determination theory and romantic relationship processes. Personality and social psychology review : an official journal of the Society for Personality and Social Psychology, Inc, 17(4), 307–324. https://doi.org/10.1177/1088868313498000 

Ryan, R. M., &amp; Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066x.55.1.68 

Deci, E. L., &amp; Ryan, R. M. (2000). &quot;The&quot; what&quot; and&quot; why&quot; of goal pursuits: Human needs and the self-determination of behavior. ''Psychological inquiry'', ''11''(4), 227–268. https://doi.org/10.1207/S15327965PLI1104_01

Slemp, G. R., Field, J. G., Ryan, R. M., Forner, V. W., Van den Broeck, A., &amp; Lewis, K. J. (2024). Interpersonal supports for basic psychological needs and their relations with motivation, well-being, and performance: A meta-analysis. Journal of personality and social psychology, 127(5), 1012–1037. https://doi.org/10.1037/pspi0000459

Van Tuin, L., Schaufeli, W. B., &amp; van Rhenen, W. (2020). The satisfaction and frustration of basic psychological needs in engaging leadership. Journal of Leadership Studies, 14(2), 6–23. https://doi.org/10.1002/jls.21695

Weinstein, N. (Ed.). (2014). Human Motivation and Interpersonal Relationships : Theory, Research, and Applications (1st ed. 2014.). Springer Netherlands. https://doi.org/10.1007/978-94-017-8542-6

}}

==External links==
* [https://www.youtube.com/watch?v=VGrcets0E6I  Promoting Motivation, Health, and Excellence: Ed Deci TEDX] (Youtube)
* [https://selfdeterminationtheory.org/theory/ Self-determination Theory] (Center for Self-determination Theory)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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      <comment>Book chapter feedback</comment>
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      <text bytes="12778" sha1="5vt5iigma4ed8sdc4ss99kj0412ob6t" xml:space="preserve">== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:|Hi [[User:{{{1}}}|{{{1}}}]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:08, 10 November 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
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&lt;!-- Title --&gt;
# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
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&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
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# Excellent – key points are well developed for each section
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
# If all planned aspects cannot be reasonably covered within the book chapter word count, be selective and concentrate on key aspects that address the question in the sub-title
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&lt;!-- Cite --&gt;
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&lt;!-- Interwiki links ---&gt;
# One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]].
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
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# Very good to excellent
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
# At least one relevant systematic review and/or meta-analysis has been identified
# What are the most relevant systematic reviews/meta-analyses about this topic?
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# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
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&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
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&lt;!-- Social contribution --&gt;
# Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making:
# One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making:
#* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 29 August 2026 (UTC)

&lt;!-- Official book chapter feedback --&gt;
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&lt;!-- Overall comments... --&gt;
# This is a very good chapter that makes very good use of psychological theory and reasonably good use of research to address a real-world phenomenon or problem
# The main areas for potential improvement are:
## reduce background theory; expand SDT in relation to dementia care
## provide more detailed review of the best psychological research about the topic
## integrate more strongly with other chapters by using embedded links to related topics
&lt;!-- Overall - Edit summaries --&gt;
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
&lt;!-- Overall – Citations --&gt;
# Excellent use of academic, peer-reviewed citations to support claims
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-determination_theory_and_dementia_care&amp;diff=2845074&amp;oldid=2841830 these copy edits]
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# Very good
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&lt;!-- Overview – Scenario - Description --&gt;
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# Clearly explains the psychological problem or phenomenon
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# The key focus question is the second half of focus question 2
# The third and fourth focus questions may be beyond the scope
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# Very good—key theories are well explained and applied; minor areas for improvements
# Reduce general theoretical background. Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question).
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# Builds on three [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
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# Good use of tables, figures, and/or lists to convey key theoretical information
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&lt;!-- Research – Key findings --&gt;
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# Insufficient use of relevant systematic reviews and/or meta-analyses
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings) 
# Claims are well referenced
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# Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
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# Good integration between the most relevant theory(ies) and the best research
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
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# Excellent summary and conclusion
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
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# Take-away messages for each focus question could be spelt out more clearly
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&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is very good
&lt;!-- Style – Language --&gt;
## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., &quot;it&quot;) instead of 1st (e.g., &quot;we&quot;) or 2nd person (e.g., &quot;you&quot;) in the main text. 1st or 2nd person can work well for case studies or feature boxes.
## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content.
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is reasonably well structured, with major sections using sub-sections
## See earlier comment about [[#Heading casing|heading casing]]
&lt;!-- Style – APA style --&gt;
# APA style
## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
&lt;!-- Style – Figures --&gt;
## Figures
### Reasonably well captioned
### The uploaded SDT diagram file name was mispelt, so I've requested renaming on Wikimedia Commons
&lt;!-- Style – Figures - Citations --&gt;
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Citations --&gt;
# Citations use good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
&lt;!-- Style – References --&gt;
## References use excellent APA style
### More copy editing is needed to fix formatting inconsistencies
|8=
&lt;!-- Learning features comments... --&gt;
# Very good use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Add more embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Reasonably good use of figure(s)
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&lt;!-- Learning features – Quizzes --&gt;
# Basic use of quiz(zes) and/or reflection question(s)
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# Excellent use of the [[{{PAGENAME}}#See also|See also]] section
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# Very good use of the [[{{PAGENAME}}#External links|External links]] section
## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] (fixed)
## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link
#|9=
&lt;!-- Social contribution comments... --&gt;
# ~17 logged, useful, mostly moderate to major/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# Note that links to academic articles should be added to the References, not embedded
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:08, 10 November 2026 (UTC)</text>
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    <title>Talk:Motivation and emotion/Book/2026/Relatedness motivation in self-determination theory</title>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Book chapter feedback</comment>
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      <text bytes="17855" sha1="5gl6dm7p6skhby0tryll381sdaf93mo" xml:space="preserve">==Potential useful sources and tips==
Hi U3203283!

I've had a read through your chapter, and I think you're developing a really interesting discussion of self-determination theory and the basic psychological needs.

I'm also doing a topic on relatedness and self-determination theory and thought you might find some of the sources I've found useful. 
* Chen, B., Vansteenkiste, M., Beyers, W., Boone, L., Deci, E. L., Van der Kaap-Deeder, J., Duriez, B., Lens, W., Matos, L., Mouratidis, A., Ryan, R. M., Sheldon, K. M., Soenens, B., Van Petegem, S., &amp; Verstuyf, J. (2015). Basic psychological need satisfaction, need frustration, and need strength across four cultures. ''Motivation and Emotion'', 39(2), 216-236. https://doi.org/10.1007/s11031-014-9450-1 
** Chen et al. (2015) examines the satisfaction and frustration of autonomy, competence, and relatedness across four different cultures. It could be useful if you're considering taking a cross-cultural perspective in your chapter.
* Vansteenkiste, M., &amp; Ryan, R. (2013). On Psychological Growth and Vulnerability: Basic Psychological Need Satisfaction and Need Frustration as a Unifying Principle. ''Journal of Psychotherapy Integration'', 23, 263-280. https://doi.org/10.1037/a0032359 
** Another source you might find useful is Vansteenkiste and Ryan (2013). They discuss the distinction between basic psychological need satisfaction and frustration within self-determination theory, including autonomy, competence, and relatedness. It could help with explaining how the satisfaction of relatedness supports more autonomous and adaptive motivation, while need frustration can contribute to less adaptive patterns of functioning. This may be useful for linking relatedness directly to differences in motivation and behaviour.

Another tip I've realised is that quite a few of the foundational and earlier sources on self-determination theory are behind paywalls. If you're connected to the university's on-campus Wi-Fi, you may be able to access some of these articles automatically through the university library's subscriptions. I've found this useful for getting access to older SDT papers that were otherwise unavailable.

Please feel free to reach out if you have anymore questions. As we're doing similarish questions, I think collaborating would help us both out!

--[[User:HawaSA|HawaSA]] ([[User talk:HawaSA|discuss]] • [[Special:Contributions/HawaSA|contribs]]) 20:12, 3 September 2026 (UTC)

== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:|Hi [[User:{{{1}}}|{{{1}}}]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:02, 13 September 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
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# Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
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# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
&lt;!-- 1-level --&gt;
# Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings)

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# Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement
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&lt;!-- Overview--&gt;
# Good
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# The scenario is promising, but adapt it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population
# The scenario figure caption could more clearly explain how it relates to the scenario
&lt;!-- Description --&gt;
# A basic description of the problem/topic is planned or presented
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&lt;!-- Key points--&gt;
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# Basic development
# It is important to communicate that SDT consists of several sub-theories, of which one is basic psychological needs theory. One of the three posited BPNs is relatedness. Put this is context and doesn't necessarily conflate with other aspects of SDT (e.g., I-E motivation).
# Good to see emphasis on satisfaction vs. frustration; this could even potentially be a focus question and dedicated section
# The topic/question doesn't call for an emphasis on wellbeing, so be wary of getting side-tracked
&lt;!-- Scope --&gt;
&lt;!-- Theory and research --&gt;
# Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples.
&lt;!-- Other --&gt;
# [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min)
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented and captioned in a basic way
&lt;!-- Caption --&gt;
# Figure caption(s) could better explain how the image connects to key points being made in the main text
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to  [[Motivation and emotion/Book|book chapters]]. Use an internal links style (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice.
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
# Place each quiz question in the most relevant section
# Focus the quiz question(s) on the take-home messages
&lt;!-- Tables --&gt;
# Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
&lt;!-- Systematic reviews --&gt;
# Well done on identifying relevant systematic reviews and/or meta-analyses
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## Basic
## Use an internal link style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- External links --&gt;
# External links
## Very good
## Use [[w:Letter case#Sentence casing|sentence casing]]
|9=
&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Excellent description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:02, 13 September 2026 (UTC)

&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is a basic chapter
&lt;!-- Overall - Edit summaries --&gt;
# Basic use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made during the writing and editing process
&lt;!-- Overall - GenAI --&gt;
# This chapter appears to be human-written
&lt;!-- Overall – Citations --&gt;
# Good use of academic, peer-reviewed citations to support claims
&lt;!-- Overall – Word count --&gt;
# Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FRelatedness_motivation_in_self-determination_theory&amp;diff=2845064&amp;oldid=2840722 these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Reasonably good
&lt;!-- Overview – Scenario --&gt;
&lt;!-- Overview – Scenario - Description --&gt;
# Engaging scenario or case study
# Ideally, use an example that a wider audience can relate to; less than 5% of the world population are university students
&lt;!-- Overview – Scenario - Figure --&gt;
# Figure 1 is relevant to the scenario
&lt;!-- Overview – Scenario - Feature box --&gt;
# The scenario uses an appropriate feature box
&lt;!-- Overview – Explains problem --&gt;
# Description of problem is too long—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections.
&lt;!-- Overview – Focus questions --&gt;
# The focus questions are reasonably good
# The focus questions seems to be unnecessarily convoluted/complicated; perhaps they were AI-generated?
|3=
&lt;!-- Theory comments... --&gt;
&lt;!-- Theory – Breadth --&gt;
# Reasonably good—relevant theories are selected, described, and applied, with some room for improvement
# Note that CET is primarily about the perception of whether feedback is interpreted as intrinsic (informing competence) or extrinsic (reward/punishment)
# Explain how CET is connected to relatedness
# Expand about relatedness support and relatedness frustration
&lt;!-- Theory – Builds on --&gt;
# Builds on ~4 [[w:|Wikipedia]] article and no related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms
&lt;!-- Theory – Depth --&gt;
# Reasonably good depth is provided about key theory(ies)
&lt;!-- Theory – Tables/Figures/Lists --&gt;
# Basic use of tables, figures, and/or lists to convey key theoretical information
&lt;!-- Theory – Citations --&gt;
# Key citations are well used
&lt;!-- Theory – Examples --&gt;
# Insufficient use of examples to illustrate theoretical concepts
|4=
&lt;!-- Research comments... --&gt;
&lt;!-- Research – Key findings --&gt;
# Basic review of relevant research
# Basic/minimal use of systematic reviews and/or meta-analyses; more emphasis would be ideal
# More detail about key studies would be ideal (e.g., sample size, cultural setting, research method, strength of findings) 
&lt;!-- Research – Critical thinking --&gt;
# Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
|5=
&lt;!-- Integration comments... --&gt;
# Basic integration between the most relevant theory(ies) and the best research
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
|6=
&lt;!-- Conclusion comments... --&gt;
# Good summary and conclusion
# Somewhat repetitive of earlier content; synthesise and summarise (e.g., specific citations aren't really needed here)
# Highlight the importance of the problem or phenomenon of interest
&lt;!-- Conclusion – Key points --&gt;
# Key points are well summarised
&lt;!-- Conclusion – Focus questions --&gt;
# Address the focus questions
&lt;!-- Conclusion – Take-home messages --&gt;
# Add practical, take-home message(s)
|7=
&lt;!-- Style comments... --&gt;
&lt;!-- Style – Written expression --&gt;
# Written expression
## The quality of written expression is reasonably good
&lt;!-- Style – Layout --&gt;
# Layout
## The headings could be more clearly aligned with the focus questions
## The chapter structure is underdeveloped
## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings
## See earlier comment about [[#Heading casing|heading casing]]
&lt;!-- Style – Grammar --&gt;
# Grammar
## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback
## Use past tense when describing research studies, although implications of findings could be in the present tense
&lt;!-- Style – Abbreviations --&gt;
## Abbreviations
### To establish an abbreviation, provide the full text followed by the abbreviation in parentheses
### Once an abbreviation is established (e.g., CET), use it consistently afterwards
&lt;!-- Style – Copy editing --&gt;
# Copy editing
## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors such as double spaces).
&lt;!-- Style – APA style --&gt;
# APA style
## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.]
&lt;!-- Style – APA style - Quotes --&gt;
## Use double (not single) quotation marks &quot;to introduce a word or phrase used ... as slang, or as an invented or coined expression&quot; ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159)
&lt;!-- Style – Figures --&gt;
## Figures
### Well captioned
&lt;!-- Style – Figures - Citations --&gt;
### Refer to each Figure at least once within the main text (e.g., &quot;(see Figure 1)&quot;)
&lt;!-- Style – Citations --&gt;
# Citations use basic [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
## If there are three or more authors, cite the first author followed by et al., then year. For example, either:
### in-text, Smith et al. (2020), or
### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020)
## Do not include author first name or initials
## Use ampersand (&amp;) inside [[w:Bracket#Parentheses|parentheses]] and &quot;and&quot; outside parentheses
## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra &amp; Aardvark, 2020)
&lt;!-- Style – References --&gt;
## The reference list is quite brief; deeper literature search is recommended
## References use reasonably good APA style:
### Check and correct use of italicisation
### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization]
### More copy editing is needed to fix formatting inconsistencies
&lt;!-- Feedback responsiveness ---&gt;
## No evidence of responsiveness to feedback on talk page
|8=
&lt;!-- Learning features comments... --&gt;
# Reasonably good use of learning features
&lt;!-- Learning features – Wikipedia embedded links --&gt;
# Basic use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
&lt;!-- Learning features – Wikiversity embedded links --&gt;
# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
&lt;!-- Learning features – Figures, tables, feature boxes, scenarios --&gt;
# Good use of figure(s)
# No use of table(s)
# No use of feature boxes other than the opening scenario
&lt;!-- Learning features – Quizzes --&gt;
# Basic use of quiz(zes) and/or reflection question(s)
# The quiz questions could be improved by being more focused on the key points and/or take-home messages
&lt;!-- Learning features – See also --&gt;
# Insufficient use of the [[{{PAGENAME}}#See also|See also]] section
## Also include links to [[Motivation and emotion/Book|related book chapters]]
&lt;!-- Learning features – External links --&gt;
# Insufficient use of the [[{{PAGENAME}}#External links|External links]] section
## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing]
## Link to the top 3 to 6 external resources about this topic
|9=
&lt;!-- Social contribution comments... --&gt;
# No logged contributions
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:44, 10 November 2026 (UTC)</text>
      <sha1>5gl6dm7p6skhby0tryll381sdaf93mo</sha1>
    </revision>
  </page>
  <page>
    <title>User:U3281277</title>
    <ns>2</ns>
    <id>331824</id>
    <revision>
      <id>2845015</id>
      <parentid>2840589</parentid>
      <timestamp>2026-11-09T20:25:31Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Social Contributions */ {{ic|Use a numbered list as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}</comment>
      <origin>2845015</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3729" sha1="pyfs2q766xylhydyfzmn5skx3c3ccsv" xml:space="preserve">==Author==

{{RoundBoxTop|theme=12}}
Hi my name is Addi and I am a third-year university student at the University of Canberra. I am currently studying a double degree of Law and Psychology, and I love both of them. I have a special interest in forensic psychology and family law and how these interact with each other within their disciplines. A big factor that influenced my interest in psychology was my high school counsellor. From my personal experience and the impact she had on me, made me want to be able to help others as much as she helped me.  

I play hockey for the UC division one team, we recently won the premiership for the second time in the last 3 years. I love this sport and have grown up playing it, the relationships and memories I have made along the way are deeply cherished and have made a significant impact on who I am today. This sport has allowed me to travel all over Australia and New Zealand and I have also had the privilege of playing international games against Hong Kong and the NZ Black sticks.   

{{RoundBoxBottom}}

==Book chapter Topic==

My book chapter topic is: 

Self-blame and emotion: How does self-blame influence emotional responses to negative events?

This chapter examines how self-blame influences emotional responses following negative events. It explores the different forms of self-blame, psychological theories that explain the relationship between self-blame and emotion, and research examining its emotional consequences. It also considers the circumstances in which self-blame may encourage positive change or contribute to ongoing distress.

==Social Contributions==

{{ic|Use a numbered list as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}}

* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAdolescent_risk-taking_and_reward-system_development&amp;diff=2840555&amp;oldid=2840074 Edited &quot;Adolescent risk-taking and reward-system development&quot;] - Corrected uppercasing to lowercasing as advised in rubric

* [https://en.wikiversity.org/wiki/Talk:Motivation_and_emotion/Book/2026/Future_orientation_and_criminal_behaviour#c-U3281277-20261015154400-Feedback_2 Commented on &quot;Future orientation and criminal behaviour&quot;] - Suggested removing in-text feedback for final submission

* [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/465940?entry_id=825801 Posted my book chapter on the discussion posts for feedback on Self-blame and emotion] - Asked for feedback on my book chapter

* [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/465940?entry_id=825802 Provided feedback on discussion post] - Gave feedback about incorporating in-text links for important topics

* [https://en.wikiversity.org/wiki/Talk:Motivation_and_emotion/Book/2026/Romantic_rejection_and_emotional_response#c-U3281277-20261015164900-Feedback Commented on &quot;Romantic rejection and emotional response&quot;] - Notified that figure 1 picture does not display

* [https://en.wikiversity.org/w/index.phptitle=Motivation_and_emotion%2FBook%2F2026%2FBoredom_and_interest&amp;diff=2840580&amp;oldid=2840304Edited &quot;Boredom and interest: Difference between revisions&quot;] - Corrected uppercasing to lowercasing as per the rubric

* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FFalling_in_love&amp;diff=2840586&amp;oldid=2839269 Corrected grammatical errors on &quot;Falling in love: Difference between revisions&quot;] - A few grammatical errors were corrected along with headings

* [https://en.wikiversity.org/wiki/Talk:Motivation_and_emotion/Book/2026/Falling_in_love#c-U3281277-20261015171900-Feedback Suggested comment on &quot;Falling in love&quot;] - Suggested formatting references in accordance with APA 7 referencing</text>
      <sha1>pyfs2q766xylhydyfzmn5skx3c3ccsv</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Motivation and emotion/Book/2026/Self-blame and emotion</title>
    <ns>1</ns>
    <id>332062</id>
    <revision>
      <id>2845022</id>
      <parentid>2842229</parentid>
      <timestamp>2026-11-09T21:34:01Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Book chapter feedback</comment>
      <origin>2845022</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16125" sha1="kyeitag514687gl2rck0zy35srie456" xml:space="preserve">== Heading casing ==
{| style=&quot;float: center; background:transparent;color:inherit;&quot;
|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:U3281277|Hi [[User:U3281277|U3281277]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:45, 15 September 2026 (UTC)
|}

&lt;!-- Official topic development feedback --&gt;
{{METF/2026
|1=
&lt;!-- Title --&gt;
# Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed)
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
# Under-developed, [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – develop further
&lt;!-- Other ---&gt;
# Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings
&lt;!-- Alignment with focus questions --&gt;
# Excellent alignment between sub-title, focus questions, and heading structure
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
# Very good
# Good
# Basic
# Insufficient
# Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- GenAI ---&gt;
# Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Scenario --&gt;
# The scenario is promising, but consider adapting it to make it more relatable to a wider international audience; university students represent only around 3% of the world's population
# A promising description of the problem/topic is planned or presented
&lt;!-- Focus questions --&gt;
&lt;!-- Alignment and quality --&gt;
# The focus questions are well aligned with the sub-title and top-level headings
&lt;!-- Other --&gt;
# Use open- rather then close-ended focus questions
# Use single- rather than double-barrelled focus questions
# Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
# Present focus questions in a feature box at the end of this section
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Promising development
# Focus on providing an integrative review of the most relevant theories and research on the topic
# I suspect, overly literal interpretation of the focus questions typical of insufficiently understood and revised genAI content
# What are the most important theories about the emotional impacts of self-blame and what is the evidence? 
&lt;!-- Scope --&gt;
# The scope is excellent (i.e., not too little/narrow or too big/broad)
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
# What are the practical, take-home messages? (address the focus questions)
|5=
&lt;!-- Figure --&gt;
# Excellent - Relevant figure(s) presented, captioned, and cited
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice.
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
# Place each quiz question in the most relevant section
# Focus the quiz question(s) on the take-home messages
&lt;!-- Tables --&gt;
# Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Very good
&lt;!-- Systematic reviews --&gt;
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## One of two links provided
### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]]
## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Use an internal links style as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
&lt;!-- External links --&gt;
# External links
## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]])
## Link to the most relevant external resources about this topic
|9=
&lt;!-- User page --&gt;
# Basic; partially sufficient
&lt;!-- Description about self --&gt;
# Very brief description about self – consider expanding
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# Add link to book chapter
|10=
&lt;!-- Social contribution --&gt;
# None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]].
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:45, 15 September 2026 (UTC)

== Resource recommendation ==
Great chapter, your discussion of how different forms of self‑blame relate to guilt, shame, and depressive symptoms is really well supported. A recent article by Fennema and Lawrence (2026) might be helpful here, as it shows that people who generalise self‑blame across many social situations are more vulnerable to depression. I have provided the reference below for you.

Fennema, D., Lawrence, A. J., Spilling, C. A., Moll, J., Lambon Ralph, M. A., &amp; Zahn, R. (2026). Self-blame-selective social conceptual overgeneralisation and vulnerability to depression. ''Cortex'', ''198'', 113–126. &lt;nowiki&gt;https://doi.org/10.1016/j.cortex.2026.02.018&lt;/nowiki&gt; [[User:StudentUC2026|StudentUC2026]] ([[User talk:StudentUC2026|discuss]] • [[Special:Contributions/StudentUC2026|contribs]]) 03:26, 23 October 2026 (UTC)

&lt;!-- Official book chapter feedback --&gt;
{{MEBF/2026
|1=
&lt;!-- Overall comments... --&gt;
# This is a very good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem
# The main area(s) for potential improvement:
#* provide more detailed review of the best psychological research about the topic
#* integrate more strongly with other chapters by using embedded links to related topics
#* express more in your own words—overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]
&lt;!-- Overall - Edit summaries --&gt;
# Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit
&lt;!-- Overall - GenAI --&gt;
# There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity
&lt;!-- Overall – Citations --&gt;
# Good use of academic, peer-reviewed citations to support claims
&lt;!-- Overall – Copy edits --&gt;
# For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-blame_and_emotion&amp;diff=2845021&amp;oldid=2841833 these copy edits]
|2=
&lt;!-- Overview – Comments... --&gt;
# Excellent
&lt;!-- Overview – Scenario --&gt;
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# Engaging scenario or case study
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# Very good—key theories are well explained and applied; minor areas for improvements
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# Reasonably good review of relevant research
# Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal
# More detail about key studies would be ideal
# In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic.
&lt;!-- Research – Critical thinking --&gt;
# Basic [[w:Critical thinking|critical thinking]] about relevant research is evident
# [[w:Critical thinking|Critical thinking]] about research could be further evidenced by:
## describing the methodology (e.g., sample, country, measures) in important studies
## considering the strength of relationships
## acknowledging limitations
## pointing out specific critiques/counterarguments
## suggesting ''specific'' directions for future research
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# Excellent integration between the most relevant theory(ies) and the best research
# Insufficient integration with related [[Motivation and emotion/Book|chapters]]
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# Excellent summary and conclusion
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# Written expression
## The quality of written expression is excellent
## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia].
&lt;!-- Style – Layout --&gt;
# Layout
## The chapter is well structured, with major sections using sub-sections
## Remove focus question numbers from headings (fixed)
## Remove colons (:) from the ends of headings
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# Grammar
## Use past tense when describing research studies, although implications of findings could be in the present tense
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## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma)
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### Reasonably well captioned
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# Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]
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# Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois
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## The number of references could be expanded
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### Check and correct use of italicisation
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# Insufficient use of learning features
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# Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]].
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# Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project.
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## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]]
## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]
## Also include links to [[Motivation and emotion/Book|related book chapters]]
## Add more links
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&lt;!-- Social contribution comments... --&gt;
# ~7 logged, useful, mostly moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]]
# ~1 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct (1-click) hyperlinks to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence.
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:33, 9 November 2026 (UTC)</text>
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  <page>
    <title>Artificial intelligence/AI content engine economics</title>
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      <contributor>
        <username>Evgenii Arsentev</username>
        <id>3110906</id>
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      <comment>Interlanguage link to the Russian version</comment>
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      <text bytes="6991" sha1="e3oodr7iu958rnsekkgvff2j5md7zl9" xml:space="preserve">{{Short description|Cost of AI-made versus human-made short video}}

This learning resource is a sub-page of [[Artificial intelligence]]. It is about a question that marketing teams and business owners now face: '''when AI agents can produce short-form video end to end, how do you compare what that costs with a human studio, with paid ads and with influencers, and what can such a comparison honestly show?'''

It works through one measured case, an AI content engine at a company under a non-disclosure agreement, and uses it to teach the method rather than to sell a conclusion.

{{RightTOC}}

== Learning objectives ==

After working through this resource a learner should be able to:

* describe what an ''AI content engine'' is and which production roles it takes over;
* compute cost per video and cost per 1,000 views from a subscription bill and view counts;
* build a staffing model for the same output when no published productivity benchmarks exist, and present the result as a range;
* explain why organic views and paid impressions are different units, and what that does to a price comparison;
* read a scenario forecast and name the assumptions that could break it.

== 1. What an AI content engine is ==

An AI content engine is a system in which AI does the work of a video studio. A language model writes and checks scripts and translations, AI avatars appear on camera, a synthetic voice does the talking, and software written by coding agents edits and publishes videos on a schedule. People set tasks and review results.

In the measured case, the engine was built by one person with no technical background, who gave a coding agent (Claude) a task in plain language and let it choose the technology. The agent wrote 99.7% of the code in the first version.

== 2. The measured case ==

Over its first 101 days the engine produced:

{| class=&quot;wikitable&quot;
! Quantity !! Value
|-
| Videos || 1,497 (2,137 language versions)
|-
| Channels || 67 on six platforms
|-
| Organic views of videos released in the period || 8,250,024
|-
| Followers and subscribers || 1,126,322
|-
| Subscriptions || about $1,300 a month; $4,313.76 over 101 days
|}

[[File:AI content engine - short-form videos produced per week (2026).png|thumb|600px|center|Videos produced per week. Output rose without new hires.]]

'''Exercise 1.''' Compute cost per 1,000 views and cost per video. ''(Answer: $4,313.76 / 8,250,024 × 1,000 = $0.52; $4,313.76 / 1,497 = $2.88.)'' Name two costs this figure leaves out.

== 3. Modelling the human alternative ==

There are no published productivity norms for short-form video, so every productivity figure below is an assumption, and the result is a range, not a point.

For 104 videos and 148 language versions a week the model assumes, for example, 3–8 scripts a day per scriptwriter and 2–5 finished videos a day per editor. With US median wages (Bureau of Labor Statistics, May 2025) and published shoot-day rates of $4,750–21,000, the model gives 42–78 people and $179,000–1.21 million a month, 138 to 935 times the subscriptions.

[[File:AI content engine - monthly cost vs human studio and influencers (2026).png|thumb|600px|center|Monthly cost of the same output (log scale).]]
[[File:AI content engine - cost per short video vs people (2026).png|thumb|600px|center|Cost of one short video (log scale).]]

Two cautions matter more than the headline:
* the model prices ''buying the same volume'' from people; it is not money any company saved, because nobody would hire that team for that volume;
* the comparison slightly favours the engine, because its cost excludes the builder's time and servers.

'''Exercise 2.''' Change the editor assumption from 2–5 to 4–8 videos a day. Which part of the monthly total moves most, and why does the range still stay above $100,000?

== 4. Ads, influencers and the unit problem ==

The company's own paid ads cost $2.23–16.57 per 1,000 impressions; sponsored influencer content typically costs $2–30 per 1,000 views. The US producer price index for internet advertising rose 34% from December 2022 to August 2026.

[[File:AI content engine - cost per 1,000 views vs ads and influencers (2026).png|thumb|600px|center|Cost per 1,000 views or impressions (log scale).]]
[[File:US internet advertising producer price index WPU366 (Dec 2022 = 100).png|thumb|600px|center|US internet advertising price index.]]

An impression, a sponsored view and an organic view of a short video are not the same unit, and the geography of the organic audience was unknown. A fair comparison therefore uses the whole range of ad prices rather than the most favourable one.

Organic reach also behaves differently in time. Paid impressions stop when spending stops; organic output grows first and views and followers catch up later. A snapshot of views understates the most recent weeks.

'''Exercise 3.''' Why would comparing the engine only with the US ad price overstate its advantage?

== 5. Reading a scenario forecast ==

The company plans to grow from 22 to 100 AI presenters. If each reached the level of its third-best, second-best or best YouTube channel, monthly views would be about 10, 25 or 178 million, against 2.5 million today.

[[File:AI content engine - forecast of monthly views with 100 AI presenters (2026).png|thumb|600px|center|Scenario forecast, not a promise.]]

The forecast rests on three fragile assumptions: that new channels grow like the best existing ones (the best channel alone brought more than three quarters of YouTube views), that growth arrives within a useful time, and that platforms do not restrict AI video.

'''Exercise 4.''' Which of the three scenarios is closest to simply scaling today's output from 22 to 100 presenters? What does that tell you about the other two?

== 6. Limitations to state every time ==

* One company, one category, one period.
* Sales are confidential, so $0.52 is the cost of attention, not of a customer.
* Some of the audience was built before the period.
* Productivity figures in the staffing model are assumptions.

== See also ==

* [[Artificial intelligence]]
* [[Large language models/Context economy of agent runs]]

== References ==

* Arsentev, E. (2026). ''AI Content Engine Economics: The Cost of Short-Form Video Production by AI Agents and by People''. Zenodo. {{doi|10.5281/zenodo.22802228}} — all figures and calculations used in this resource.
* Teutloff, O. et al. (2025). Winners and losers of generative AI: Early evidence of shifts in freelancer demand. ''Journal of Economic Behavior &amp; Organization'' 235:106845. {{doi|10.1016/j.jebo.2024.106845}}
* Demirci, O., Hannane, J., Zhu, X. (2025). Who is AI replacing? The impact of generative AI on online freelancing platforms. ''Management Science'' 71(10):8097–8108. {{doi|10.1287/mnsc.2024.05420}}

[[Category:Artificial intelligence]]
[[Category:Economics]]
[[Category:Marketing]]
[[ru:Экономика ИИ-контент-завода (учебный кейс)]]</text>
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  <page>
    <title>Social Victorians/Boxing/Smith-Slavin 1889-12-23</title>
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      <text bytes="188852" sha1="sytkywa4u76kohpauctijxh0pcr5jf2" xml:space="preserve">=Event=
International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium.

==Overview==
On 23 December 1889 a championship boxing match in Bruges, Belgium, ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin or getting him near the ropes and beat him up, using fists, sticks, &quot;knuckle-dusters,&quot; &quot;life preservers&quot; and boots. The referee ruled the match a draw to prevent Slavin from being killed and then fled. The ''Cork Constitution'' called it &quot;the Smith-Slavin piece of brutality.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;

A knuckle-duster was &quot;A metal instrument made to cover the knuckles,&quot;&lt;ref&gt;“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.&lt;/ref&gt; like brass knuckles, and a life preserver was &quot;A stick or truncheon weighted at one end&quot; with lead.&lt;ref&gt;“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.&lt;/ref&gt;

One meeting of the boxing-oriented London Pelican Club was held almost immediately. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below).

Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England.

Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made here to offer complete coverage. Specialized sports coverage is privileged.

==Logistics==
* 23 December 1889, Monday, 8:00–9:00 a.m., actual fighting commencing at 8:35
* &quot;the private grounds of a gentleman's residence near Bruges, Belgium&quot;&lt;ref name=&quot;:1&quot;&gt;&quot;The Great Fight. Smith &amp; Slavin for £1000. Disorderly Proceedings. Brutal Attack on Slavin. Smith Knocked Down Repeatedly. He Leaves the Ring. A Draw Declared after 14 Rounds.&quot; ''The Sportsman'' 24 December 1889, Tuesday: 3 [of 4], Col. 1a – 4, Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&amp;page=3&amp;article=057&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; (3, Col. 1a)
* Stakes: £1,000
==Related Events==
* [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]]
* the &quot;[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]&quot; 
==Who Was Present==
# Jem Smith, British boxer
## Jack Harper, Smith's second
## Jack Baldock, Smith's second
## Mr J. Fleming, umpired for Smith
# Frank Slavin, Australian boxer
## Josh Cosnett, Slavin's second
## Bob Kirby, Slavin's second
## Mick Taylor, &quot;superintended the arrangements in his [Slavin's] corner&quot;
## Mr W. Riley umpired for Slavin
# &quot;the company round the ring
## Lord Mandeville — [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]]
## Lord Royston
## Hon. M. Sandys
## Mr Abington — Mr. G. Abington Baird, Smith's manager? and horseman (see description in Anthology)
## Mr F. Kotchie
## Mr J. Kilby — Jack Kilby, &quot;a real sportsman&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## Mr C. Parsons
## Mr Ike White
## Mr C. Morton
## Mr Harris
## Mr E. Bayley — Teddy Bayley
## Mr John Lewis (Australia) — Jack Lewis, Slavin's manager
## Mr J. Vesey — Joe Vesey, more often, or Vezey, referee
## Mr F. Hinde — Frank Hinde
## Mr M. Taylor
## Mr J. Calder — timekeeper
## &quot;Pop&quot; Joy
## R. Habbijam — Bob Habbijam
## G. (&quot;Barney&quot;) Sheppard — &quot;superintended the pitching of the ring&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3b)
## Jem Carney
## Tom Smith
## Mangie [?] Hyams
## Matthews
## Summers
## Boulton
## Mr G. T. Dunning — Tom Dunning, worked at ''The Sportsman'' and served as treasurer for the subscription to collect a cash prize for Slavin.
## &quot;several others, including two or three reporters&quot; — &quot;three scribes with pockets bulging from a surplusage of note-books dropped in&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## The writer of the big 24 December article, who reports for ''The Sportsman'' and who is not the &quot;well-known sporting journalist&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1b) who had heard that Slavin had been arrested in Margate
# &quot;an English resident in Bruges&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3a)
# &quot;Pop&quot; Hinde
# C. Horton
# &quot;a mob of Birmingham men&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
# Lord Rosslyn ?
# &quot;The Squire&quot;
#
==Entertainment==

==Anthology==

=== The Sportsman ===
The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 and most of page 4 (nearly 13 full columns) entirely to this story. This reporter's writing style is very ornate:

* &quot;Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping&quot; (3, Col. 1c)
* &quot;For a time it [the sea] was a bit rough, and one of the good sailors commenced quoting Byron. 'O'er the glad waters of the dark blue sea,' sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2a)
* &quot;Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2b)
* &quot;While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
* &quot;He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)

Besides the street-fighting terms (knuckle-dusters, life preservers) above, he also uses slang and local vocabulary like sabots (wooden clogs worn by working-class Belgians), Brums (people from Birmingham [?])

This is, then, a very long article, reproduced here in full except where the text is illegible, the very large block quote broken up by level-2 heads for navigation:&lt;blockquote&gt;The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below.&lt;/blockquote&gt;

==== THE JOURNEY TO THE CONTINENT. ====
&lt;blockquote&gt;(By our special reporter.)

[?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of &quot;globe-trotting,&quot; as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came]

A Rude Shock.

[Arriving??] at the office of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that &quot;something had happened.&quot; His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. &quot;What do you think of this?&quot; asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. &quot;Stoppage, but think can get bail. — Slavin.&quot; Here was a pretty &quot;go,&quot; the mystic words being translated meaning that the Australian had been &quot;pinched&quot; at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it

A May Morning.

Long before '''the earliest purveyor of the lacteal fluid''' arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of &quot;London peculiar&quot; had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking &quot;all broke up,&quot; put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a '''surplusage of note-books''' dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, &quot;Pop,&quot; and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey [sic], who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross &quot;The Squire,&quot; Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''.

Down to Dover.

It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly &quot;gave the game away,&quot; and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as &quot;dirty&quot; weather in store.

On the Briny.

The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and &quot;once on board the lugger&quot; my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the &quot;Princess&quot; swung off from the pier the two &quot;Bobs,&quot; Kirby and Habbijam, betook themselves to the &quot;rail&quot; in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. &quot;O'er the glad waters of the dark blue sea,&quot; sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having &quot;sought the seclusion which the cabin grants,&quot; and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in &quot;Pop,&quot; and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. [3, Col. 2a–2b] 

Ostend to Bruges.

While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded &quot;foreigner&quot; who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, &quot;Well, when is the 'mill' coming off?&quot; In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any &quot;mills.&quot; As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring '''that useful saponaceous compound''' with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity. [3, Col. 2b–2c]

Pugilism at the Piano.

While '''awaiting the refection''' Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The &quot;pelting of the pitiless storm&quot; continued the greater part of the night. 

Arrival of Jem Smith. 

The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arrangements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or &quot;the manger&quot; as &quot;Pop&quot; expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery &quot;Good morning, gentlemen,&quot; Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. 

The March to the Battlefield. 

Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out again. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work [3, Col. 3a–3b] 

Building the Ring. 

Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. 

The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, &quot;Pop&quot; Joy, R. Habbijam, G. (&quot;Barney&quot;) Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that 

The Dreaded Birmingham Division 

had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the &quot;boys&quot; worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, &quot;I hope all those present will keep quiet and help me all you can,&quot; at which brief speech there was a chorus of &quot;hear, hear,&quot; especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. 

The Call of &quot;Time&quot; 

At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, 

Smith was Knocked Down 

By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their &quot;Champion&quot; did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the &quot;Champion&quot; wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to &quot;be a man.&quot; Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of &quot;Here they are&quot; was raised, and 

A Stampede Ensued 

from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, &quot;There they are, there they are.&quot; By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, &quot;Are you Englishmen? I am an Englishman, and only want fair play.&quot; The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a &quot;duster,&quot; the referee saying, &quot;I saw that.&quot; As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, &quot;I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play.&quot; The referee concluded by giving the decision, &quot;It's a draw.&quot; The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting &quot;There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance.&quot; There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. 

There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended 

Making All Speed Homewards. 

The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. 

Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. &lt;/blockquote&gt;

==== THE FIGHT. ====
&lt;blockquote&gt;First Round. 

Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. 

Second Round. 

No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, &quot;First blood,&quot; a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On &quot;Time&quot; being called the pair were quick to scratch, that is to say, Slavin walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec.

Third Round.

Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking back to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec.

Fourth Round.

Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, 1min 15sec.

Fifth Round.

&quot;Don't go into his corner,&quot; urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite corner, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds.

Sixth Round.

Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec.

Seventh Round.

In vain did Slavin's friends implore him not to go into Smith's very &quot;warm corner.&quot; It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion was something terrible, and the ring looked very much like being invaded. The ropes in Smith's corner were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. 

Eighth Round.

Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth. Time of round, 1min 30sec.

Ninth Round.

This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a]

Tenth Round.

The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. 

Eleventh Round. 

Neither was quite so quick to time, and Harper, noticing that, said, &quot;This is not a two-round job, Patsey.&quot; Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to &quot;Come on and have a round.&quot; This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. 

Twelfth Round. 

A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. 

Thirteenth Round. 

Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, &quot;Oh!&quot; Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of &quot;Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a &quot;blue.&quot; The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words &quot;fight on&quot; came he and his friends were very delighted. 

Fourteenth and Last Round. 

When &quot;Time&quot; was called Slavin folded his arms and waited for Smith to &quot;come on,&quot; but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. 

No sooner was this round over than the referee called out. &quot;I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw.&quot; Then, going up to Slavin he said, &quot;I will not stay to see you killed.&quot; The Antipodean stood up and exclaimed, &quot;If you are Englishmen why don't you give me fair play? I have not had fair play.&quot; 

After this came a stampede. 

Remarks. 

Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as &quot;fights in the old style.&quot; That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the &quot;upper crust.&quot; Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being &quot;bashed&quot; on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with &quot;dusters&quot; and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a &quot;glover.&quot; With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a] that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or &quot;bottom&quot; of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, &quot;he was just getting warm.&quot; Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect.  

For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved.

From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play.

Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow. Bearing on this subject, Rule 26 says:— &quot;That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle.&quot; Rule 24 is also plain enough in all conscience, &quot;That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the &quot;boys&quot; and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but &quot;Kim&quot; soon showed he could hold his own with the &quot;boy,&quot; who beat a hasty retreat at the &quot;sight&quot; of six-inch &quot;Jack.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 1a)

We now take our leave of this unsavoury affair, which will probably be the last of its kind — &quot;a Championship knuckle fight.&quot; The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner &quot;merry mills in the old style&quot; die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office about three o'clock.

Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of &quot;dusters&quot; the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye.

It may be stated that not a single bet was made at the ring side. [4, Col. 1b]&lt;/blockquote&gt;

==== SLAVIN'S OPINION OF HIS TREATENT. ====
&lt;blockquote&gt;His Future Plans. 

On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. 

Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. 

With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. 

Much sympathy was expressed on all sides, and in the course of conversation with a S&lt;small&gt;PORTSMAN&lt;/small&gt; representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] 

Did you expect any trouble at the ring side? 

Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. 

Do you feel any effects of the fighting? 

Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. 

Why did you persist in going to his corner? 

Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. 

Of course you heard a mob was got together for Smith? 

Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. 

What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. [4, Col. 1c] 

I suppose you have a poor opinion of English fair play? 

Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. 

What are your plans for the future? 

Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. 

With the gloves? 

Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. 

You prefer the old style to the new? 

Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. 

Shall you give any exhibitions? 

Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. 

You put on weight the last week? 

Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. 

What do you think of our seconds? 

Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. 

When do you propose visiting London? 

Slavin: I'll come up to-morrow and call at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office about three. 

A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. &lt;/blockquote&gt;

==== A Testimonial for Slavin. ====
&lt;blockquote&gt;So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.'' The maximum donation was 2 guineas, which would be 2£ 2''s.''] 

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Hon. Michael Sandys  2 2 0
* Mr Fred Ketchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Mr J. Calder  2 2 0

Any contributions sent to the hon. treasurer will be duly acknowledged in T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;, which in a couple of days will contain the names of a committee now in course of formation. 

In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; 

Sir, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  &quot;Champion of England&quot; to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. 

If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C. 

Royal Thames Yacht Club, December 23. 

[We refer our correspondent to the paragraph above. — Ed. T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. ] [sic bracketed interpolation]  [4, Col. 2b] &lt;/blockquote&gt;

==== Eyewitnesses Interviewed. ====
&lt;blockquote&gt;A representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known &quot;sports&quot; (including more then one scion of a noble house) for a few minutes' chat. 

Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which did not number more than forty, being particularly noticeable. 

&quot;How did the combatants' seconds behave?&quot; — &quot;Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes.&quot; 

&quot;What form was the Englishman displaying?&quot; — &quot;From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds. Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy.&quot; 

&quot;How do you account for such a noisy corner at the back of Smith?&quot; — &quot;Although they did not number more than eight, they were some of the worst characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, inasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, &quot;framed&quot; [sic double quotes] up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his party.&quot; 

Are these men known at all?&quot; — &quot;Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period.&quot; 

&quot;Was not this sort of trouble expected?&quot; — &quot;It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated.&quot; 

In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. 

&quot;How did the crowd behave towards the finish?&quot; — &quot;There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious.&quot; 

&quot;Is it true that attempts were made to tamper with the water used by Slavin's seconds?&quot; — &quot;Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail.&quot; 

&quot;Did Smith deal out much punishment?&quot; — &quot;His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought for two hours more. He could have beaten Smith and both his seconds into the bargain, given fair play.&quot; 

&quot;Was Slavin injured during the fight?&quot; — &quot;No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed.&quot; 

&quot;As to the referee's decision?&quot; — &quot;That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened.&quot; 

&quot;What becamiof Smith after the fight.&quot; — &quot;He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does.&quot; 

&quot;How did the Cornstalk behave in the ring?&quot; — &quot;He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion.&quot; 

Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a &quot;mill,&quot; was very severe in his comments on the gang who were so lost to all sense of decency and fair play. 

Mr White said: — &quot;The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He had to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful.&quot;  [4, Col. 2c] 

&quot;Do you think it was for the best that the contest was stopped as it was?&quot; 

&quot;Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]&quot; 

&quot;What do you think of Slavin's abilities?&quot; 

&quot;He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt.&quot; 

On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. &quot;He is coming to town to-morrow,&quot; said the speaker, &quot;and shall have a great reception. Some good sportsmen have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it.&quot; 

&quot;What seemed the general opinion on the ground?&quot; 

&quot;Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took.&quot; &lt;/blockquote&gt;

==== Reception of the News in Birmingham. ====
&lt;blockquote&gt;Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a] and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. 

How the Match Was Made 

On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— &quot;He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, was unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers.&quot; [sic: quotation begins at the top of the paragraph] 

Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; and signed:   

Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', for the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally.   

In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names.  [4, Col. 3c]   

Frank Hinde (for Frank Slavin).   

John Fleming (for James Smith).   

Witness for Smith, F. Montague.   

Witness for Stavin, John Lewes.   

Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The &quot;Cornstalk,&quot; though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not step in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds.   

Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell.   

Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[,] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice.   &lt;/blockquote&gt;

==== The Men in Training ====
&lt;blockquote&gt;Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp.   

Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound,  Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson.   &lt;/blockquote&gt;

==== Finding the Battle-Ground. ====
&lt;blockquote&gt;(By one of the Explorers.)   

At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class &quot;smoker&quot; attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed that a &quot;choppy sea&quot; was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. &quot;How confoundedly annoying,&quot;  he said. &quot;This is the sixth time I have been across during the year, and not once has it been calm.&quot; &quot;Better have a sleeping berth, sir,&quot; said a sprightly youth at the top of the stairs. &quot;Sleeping berth be hanged. I'll remain on deck,&quot; said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit &quot;peckish&quot; and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the &quot;magging&quot; [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, &quot;struck ile [?]&quot; with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The &quot;elderly party&quot; had a voice greatly resembling the dividing of tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. Just as I thought of giving up sleep for a bad job a gale sprang up. Though the Geselle is a record breaker she soon felt the effects of the breeze and began to roll heavily. The increase in the motion of the steamer was followed by a decrease in the conversation around me, and by degrees the punster went from had to worse. The &quot;[hussey??]&quot; [one?] also dropped into a walk, and after monosyllables had been exchanged the two came to a dead stop. Offering up a prayer of thanksgiving for this I closed my eyes again, and being a good sailor was quickly sleeping the peace of, I hope, the just. The concussion of the Gazelle against the Guernsey Pier, accompanied by a prod from Fleming's stick, at length aroused me.  [4, Col. 4b]   

Among the Channel Islands.   

The time by &quot;Joe Vickers&quot; was 7 a.m., and half-awake I took to my boots. &quot;Need not lace them; we shall be there in a minute,&quot; said Fleming, and in a drowsy, submissive, take-me-anywhere-if-only-to-bed sort of way I followed him on deck. A Babel of voices and a clutch at my bag and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a &quot;[sheenie?]&quot; (goodness knows what he meant, though), made us welcome, and agitating the communicator he handed us over to the care of a dark-eyed chambermaid. I forget her name, and as Fleming is not here I cannot give it. I will say no more than that after a cup of tea diluted with delicious cream, we took a well-earned and mucch needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the &quot;holiday maker&quot; and the &quot;honeymooner,&quot; is, as everybody knows, or should know, situate in the English Channel near the coasts of Brittany and Normandy. It is 68 [?] miles from Weymouth, 7 from Sark, 19 from Alderney [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in &quot;Mike.&quot; Said I, &quot;He's a Frenchman.&quot; &quot;Get out,&quot; replied Fleming, &quot;he's a 'bark.'&quot; &quot;Drinks he isn't,&quot; &quot;Done; drinks he is.&quot; &quot;Mike&quot; had not at that time spoken, but had noiselessly passed to and fro with a military step and a hand behind him. In reply to a call from Fleming he approached, and a &quot;Yes, yer honour,&quot; told me I should have to &quot;part.&quot; &quot;Mike&quot; was a trump, and I loved him from the start. For eleven years he had been in the service of a colonel at 1''l.'' a week, and three suit of clothes in the year. Hotel pay was not so good, and though he made &quot;a bit&quot; in the summer, the young ones were generally there when the bills were paid, if not when the work was on. &quot;Mike&quot; told us lots of things in what many consider the &quot;fascinating brogue.&quot; He informed us — and how our hearts beat at the news — that the island, which had a population of over 32,000, employed but six policemen. &quot;There's not a beggar in the land,&quot; said Mike, &quot;and but one Hansom cab, and,&quot; pointing to the window, &quot;be jabers, there it goes. &quot; So far so good, but after a walk round the son of Erin gave us a shock by saying, in reply to a query from Fleming, &quot;They've a way of their own on this island, and nearly ivery [sic] other man is a special constable.&quot; Fortunately we had just finished our repast, and, supporting each other, we tottered faintly to the door, partially regaining strength with a breath of the pure sea breeze. Beads of perspiration stood on the brow of the Pelican, whilst I was suffering from a &quot;trickling&quot; down the back of anything but a pleasing nature. Visions of a dungeon-cell under Castle Coronet flashed through my brain as a voice and a step were heard behind, and then I was somewhat reassured. &quot;When would you like a chair?&quot; fell softly on mine ear. Afraid that we had been thought of as a bit week on the leg, I turned to the speaker and asked what he meant. Explanations ensuing, we found the &quot;chair&quot; was a sort of Bath chair on wheels used throughout the island. Declining the offer, we took a stroll, and putting this and that together, came to the conclusion that our task was of a very delicate nature. The first thing was to interview the captain, who, in reply to a note, courteously waited upon us. He gave us the geography of the island and an introduction to Mr M., a friend, who we unfortunately found away from home. Taking a &quot;chair&quot; we then started on a voyage of exploration, making, as advised, our way to L'Ancresse [?] Common. Leaving St. Peter's Port and passing through St. Sampson's — the Vale — we came across acres of vineyards, in one instance extending nearly a mile along the road-side. Half an hour's drive brought us to the common mentioned, and dismounting we tried in vain to find a level piece of turf. Windmills abounded on all sides, and plodding on we reached L'Ancresse Bay, a lovely little spot, where the sea rippled softly over a bed of almost snow-white sand. Nowhere could be found a suitable spot whereon to pitch the &quot;magic square,&quot; and as the shades of night were falling fast we rattled back to the &quot;Royal.&quot; A wet night offered no inducement for further travel, and it was a case of early to bed and early to rise, the next day finding Fleming and myself at the door of Mr M. Acquainting this gentleman with the Nature of the business in hand he, like a good sportsman, fell into the scheme and at once volunteered to drive us to &quot;the best spot in the world.&quot; &quot;Bessie&quot; was harnessed to the village cart, and away we went at the rate of fully fourteen miles an hour in the direction of the Forest. I have sat behind many a good goer, but must for her size award the palm to Bessie a beauty barely twelve hands high. Throwing the reins in the bottom of the vehicle, Mr M. let the pony have her head, and the way she went uphill, down dale, and past other traps was truly astonishing. Great St. Peter's was passed, and lovely scenery encountered, the gardens being full of flowers, geraniums and roses blooming on all sides. An enjoyable drive of forty-five minutes found us at Boequaire [?] Bay, and directly in front stood the island of Lihou. As many of our readers may know, Lihou is a small hill of land that at low water can be reached on foot from Guernsey. As a fact, a paved roadway is laid in the bed of the ocean, across which both foot passengers and carts easily make their way. With the tide up communication is effected by boats, but this being low at the time we had the novel experience of   

Driving Through the Sea.   

Gaining the other side we &quot;explored&quot; and as the piece of land is only some three hundred yards by a hundred and fifty, we were not long about it. Two excellent patches of level turf were found, but just as we were making for the trap a gaunt, Robinson Crusoe-looking individual scaled one of the small stone walls and barred the way. Of course, he wanted to know our business, and, after a rather rough opening, quickly &quot;listened to reason.&quot; In the course of conversation we found that the population of the island was limited to three, the old &quot;josser&quot; and his two daughters, with a &quot;moke&quot; thrown in. Pointing a long skinny finger, which might have belonged to the Witch of Endor, towards the quadruped mentioned, the old man said, &quot;Us two keeps the family. He draws the seaweed and I burn it.&quot;  &quot;And why do you burn it?&quot; &quot;Well, to make [Col. 4c–5a] ashes, and from the ashes iodine is manufactured.&quot; &quot;And how much do yen earn a week by that?&quot; said  I. &quot;Oh, about 4''s.''; not 5''s.'' any week.&quot; Great Scott! and this was all the old man, his two daughters,  and the &quot;moke&quot; had to live on. Handing him a franc each, which nearly stupefied the old chappie, Fleming and myself returned to Mr M., who drove us rapidly back to St. Peter's Port. The next day we crossed to Sark and Jersey, but found nothing so suitable as the ground at Lihou, and fairly well satisfied with the result of our visit, a day later boarded the Lynx for the return journey.   

During our visit the weather had been of a fairly favourable description, but the voyage home was made under very depressing circumstances. Rain fell heavily and a thick mist hung over the water, which after going a few miles was alive with porpoise. Some of these animals came within a few yards of the Lynx, but the speed of the boat proved too good for the finney ones, who were eventually left behind. With a breeze springing up sail was hoisted, and dashing through a heavy sea in splendid form the Lynx made some sixteen knots in the hour. In this way half the journey was completed, when all of a sudden we were caught in a squall. In an instant the wind chopped round, and before you could say &quot;Jack Robinson&quot; the good vessel was turned broadside to. The order to take in sail was promptly obeyed, but the clatter on deck brought many an anxious passenger rapidly from below. A very heavy sea was now running, and though the Lynx kept her course bravely the decks were swept from stem to stern. A huge East India-man now hove in sight, and, as one of the men informed me, she had the audacity to cross our bows instead of passing under our stern. This breach of ocean etiquette ruffled the skipper of the Lynx, who chased the trader, and eventually made her alter her course. This little &quot;diversion&quot; caused much amusement amongst our passengers, and after a cold, rough passage we landed safely at Weymouth. Chilled to the bone, we made for the Victoria Hotel, where a chop and a glass of old October revived the almost stagnant circulation. London was reached about 9.30, and, tired and worn out, I was glad to lay my head on a pillow I had not seen for four days.   

From London to Ostend.

Unforseen [sic] difficulties having cropped up, the Channel Island scheme had to be abandoned; consequently we were once more called upon to leave our native land. This time Belgium was our destination, and having carefully studied the map of that country, we took our departure one fine starlight evening from Charing-cross by the 8.15 Continental express. In the previous week we had the luxury of a compartment to ourselves, but on the occasion under notice we were not so fortunate. Two individuals, one with a strong Midland accent, and the other a Frenchman speaking bad English, shared the carriage with us, and at such a rate did the former's tongue go directly he entered that I gave up reading as a bad job, with a sigh put away my unfinished book of the previous trip. Fleming fared better, and having a knack of ignoring all around him, once more tackled &quot;Blood Money,&quot; and almost read himself off to sleep.  In the meantime our friends in the opposite corner made &quot;hay,&quot; and I soon found out they were both in the lace trade. With a tongue that went a couple of hundred to the minute, the Midlander almost monopolised the conversation, from which I gleaned the two were bound for Calais, and thanked goodness we should part at Dover.

Reaching this port about ten we boarded the Ostend boat, which, as regards passengers, was very poorly patronised. Hardly a breath of wind was stirring, and with the sea calm as a mill-pond we had a pleasant passage over, and landed safely at Ostend soon after 3 a.m. A cup of coffee was hastily swallowed, and a half-hour's run in the Brussels express found us  [4, Col. 5b]

At Bruges.

Leaving the train, we had considerable difficulty in finding our way out of the station. For a moment we caught sight of a drowsy old porter, but the last carriage had not passed away from the platform ere he was lost to view. Seeing an opening on other side we made for it, only to have our passage barred by a half-dressed Belgian who, whilst displaying a total ignorance of English, Irish, and Scotch, directed us to return the way we came. For some time we tried in vain to find an outlet, and just as I had made up my mind to walk the platform until daylight Fleming found an iron gate ajar. Without loss of time we were in the street, and crossing the round, knocked loudly at the door of the nearest hotel. All was in darkness, and with no response to our fourth appeal for admittance, a return to the station or a &quot;tramp round&quot; appeared to be our only alternative. A slight noise inside followed by a shuffling of feet, again raised hope within us, and when, after some trouble, the door swung back and brough to view a shock-headed Belgian our relief was great. No time, I can assure you, was lost in getting between the sheets, and rising betimes we again went on a voyage of discovery. Fleming had a chum in the town, and to his house of call our steps were directed. Again disappointment was in store for our wanderers, and a feeling of disappointment came over me when we found Mr G. was away. There was, under the circumstances, nothing left but to wait, and Fleming, who puts on hair rapidly, decided to place himself under the care of the town barber, or, I should, properly speaking, my ''coiffeur''. For this piece of indiscretion he had to come away minus a slice of his lower lip, which, fortunately, was not large enough to seriously reduce his avordupois [sic?], and heaping blessings (?) [sic] on mankind generally, not in any way forgetting the &quot;Bruges Barber, we roamed over the old town, noted for its magnificent cathedral and many seats of learning. The hours sped on pleasantly enough, and in due time our friend was unearthed. Being pledged to secrecy I am not going from my word by giving full particulars of

The Battle Ground

placed at our disposal. This is in close proximity to the town — in fact, not half an hour's walk, and on reaching the spot we were shown a lovely piece of turn. A clump of trees on each side partly hid the house from view, and when I say that the ring was to be pitched on a lawn tennis ground, it will be seen that neither side could claim an advantage as regards standing ground, &amp;c., the lawn being as level as a billiard table. Though our efforts so far had been crowned with success, we did not get permission to use the &quot;pictch&quot; without much trouble. The matter had to be seriously considered, and Mr G. would not give a decided answer without waiting another day. Consequently our stay had to be extended and the following morning, after repeated promises that the affair should be conducted in an orderly manner with not a rough at the ring-side, we got the answer, &quot;Yes.&quot; &quot;But,&quot; said Mr G., &quot;if there is any row or crowd I can assure you you will have a journey for nothing; therefore if trouble is at all probable keep away and save your money and time.&quot; As long as faith was not broken by either side we saw no reason why the matter should not be settled in a proper manner, and making all the necessary arrangements and laying down plans of a most complete description, we returned to town well satisfied with the result of a visit to Bruges.  [4, Col. 5c]

Trouble about the Referee. 

On returning to town we received satisfactory reports of the work done by each of the contestants. Slavin, looked after by Josh Cosnett, the Birmingham heavy-weight, and his friend Mr Michael Taylor, of Australia, though rather light, was in good health. At one time he scaled only 12st 8lb, but a judicious spell of light work brought him back to 13st odd. Considering the importance of the business in hand, and its bearing on future events, Slavin had some  cause for anxiety. In addition to this he was fighting for 300''l.'' of his own money, and consequently wished to do himself justice in the international battle. 

The next question was the selection of someone capable and willing to fill the all-important post of referee. According to the articles it was left to the stakeholder to appoint the official named providing the representatives failed to agree, and in the end this course had to be adopted, solely owing to the objections lodged by Smith. The first gentleman suggested by Slavin's side was Mr B. J. Angle, whom Smith promptly objected to. Next a well-known owner of racehorses (a gentleman who adjudicated upon the knuckle fight between George Camp and &quot;Click&quot; Soles) was proposed with no better result. Mr Bat Murphy, strange to say, would not suit, and then in succession &quot;No&quot; was said to the names of  Messrs G. H. Vise, A. F. Bettinson, and [B.?] Bayley. All these gentlemen, it must be stated, were proposed by Slavin and objected to by Smith, who in the meantime had named a member of the Pelican Club, a gentleman of limited experience in these matters, and the landlord of the house from which he was training, Mr W. Riley, Newmarket. This brought about a deadlock, and the state of affairs getting wind, ugly rumours were soon afloat, and even money, and in many cases good odds, were betted that there would be &quot;no fight.&quot; All this while Slavin was chafing and fuming at the delay, whilst at the same time his  representative and Smith's were at work almost night and day endeavouring to bring about a settlement of the difficulty. For a long time it looked as though the matter would end in smoke, as after the services of the stakeholder had been called upon great difficulty was experienced in getting anyone to accept the office of referee. So late as Friday at noon nothing had been done, and it was not until the evening of that day that Mr Joe Vesey, consenting to stand, was nominated by the stakeholder. 

A Scare.

Though the battle-ground was only known to two persons, the country where the fight was to be decided being, in fact, or supposed to be, a perfect secret, on Friday morning a paragraph said to have been sent out by a news agency appeared in several papers, and created intense astonishment and consternation. This, set out in large type by those who we should have thought would have suppressed the account, stated that Smith on the previous evening had arrived in London from his training quarters, and even went so far as to name the street he slept in. That Smith's visit to town was extremely injudicious there is not the slightest doubt, but this was not the worst. The time of departure for the scene of action was given, with a description of the place selected. This publicity had the effect of keeping away many who bad expressed a wish to be present, and on Slavin's side no less than eight gentlemen, each holding a high position in the sporting world, declined to have anything further to do with the affair. Who divulged the secret is at present a mystery, but there is not the slightest doubt that the information so freely put forward not only opened the eyes of the authorities, but gave the &quot;mob,&quot; to a certain extent, a key to the [Col. 5c–6a] situation. Slavin received the news with no small amount of astonishment, and a couple of telegrams informing him that a batch of the &quot;boys&quot; had gone across to make things warm for him did not give peace to his troubled mind. Smith made his way out of England on Friday by the short sea route, Dover to Calais, which he reached in safety. Slavin, not so fortunate, was arrested on Satuday, as already reported, but, after being bound over to keep the peace, he got away the same evening.&lt;/blockquote&gt;

==== Performance of the Men. ====
&lt;blockquote&gt;Jem Smith

was born in Whitecross-street, in the parish of Cripple-gate, on January 21, 1863. He stands 5ft 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in high, and weighs in condition 12st 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;?&lt;/sub&gt;in; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; hips, 41in; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; calf, 16&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing, after one or two rough-and-tumbles with his mates he was taken in hand by Jack Knifton (&quot;81-tonner&quot;), and later by Mr F. Grimm, of the Central Club. Smith has never been defeated, though in his first engagement he did not shape like a coming Champion. His first success was in a competition at the Griffin, Shoreditch. Later he beat Snowey, of Holloway, and was then matched to box Harry Arnold, with gloves, for a 10''l.'' purse. Arnold, though a stone and a half lighter than Smith, stripped in good condition, having for some time previously been engaged in training Jack Massey for an important engagement. The affair was looked upon as a good thing for Smith, but it took him nearly an hour to polish off the Seven Dials' representative, though he had slightly the best of the encounter from start to finish. Subsequently the veteran Goode deposited a [sov?] on behalf of Smith to match him to fight Massey, but this was forfeited. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for 25''l.'' a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end twelve rounds. Smith also won Tom Symonds's All-comers' Competition on February 26[?], 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. At the close Smith was presented with a large oil painting of himself, executed by a prominent patron of the sport. Smith's next battle was with Jack Davis for 100''l.'' a-side and the Championship, on the turf, and came off on the borders of Surrey and Sussex, on Wednesday, December 17. In all six rounds were fought, occupying 7min 56sec, Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for 300''l.'' at Maisonne-Lafitte, near Paris, on Tuesday, February 16, 1886 [?]. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Jack Knifton, the &quot;81-tonner,&quot; as he is called. This proved a ''fiasco'', a two days' dreary tramp on the Continent being followed by a return to the old country. Knifton's people claimed that Smith had more than the specified number of followers, but there is little doubt that only one side wanted a fight, and that was not the &quot;81-tonner's&quot; lot. Meeting after meeting was held, Smith offering to fight anywhere the other liked, and after much wrangling arrangements were made to bring the affair off near Oxted [?]. Somehow of other the police got wind of it, and overtaking the Knifton party near Croydon rendered another postponement necessary. this time the arrangements were left in the hands of Knifton's side, with the result that the principals, seconds, and a dozen others were captured early one morning in a furniture van by the police before they had gone fifty yards as their journey. All were brought up at Thames Police-court, and after a remand bound over to keep the peace for twelve months. The memorial struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19, 1887 [?]. Jem's recent performance was a Ten-round Glove Contest with Jack Wannop, the Champion laying 250''l.'' to 200''l.,'' and winning easily, following which he was beaten in two rounds at the Pelican Club by Peter Jackson, the Coloured Champion, in a glove contest, for 1000''l.''  [4, Col. 6b] 

Frank P. Slavin. 

the Champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft 1in in his stockings, and in condition scales 13st 5lb or 13st 6lb. Slavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. He always had a liking for the gloves, and sparred a great deal when young, but did not enter into any match until the latter part of 1885 — that is, for money, though he won several competitions previously as an amateur. Slavin's first fight was with knuckles against M. Power for 50''l.'' a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with 2oz gloves for 200''l.'' a-side and the Championship of Queensland, winning in three rounds. This was in December, 1885 [?], and in the following year the Australian formed a combination and went touring, offering purses ranging from 10''l.'' to 20''l.'' to any man who stood up for four rounds. At this game Slavin had an easy time of it; some stood one round, and some two, but none of them four. Towards the end of the year he went to Sydney, and entered into an agreement with Larry Foley, who has a big place there. Here Slavin met Taylor, a pupil of Jackson, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big fellow named Bligh for 10''l.'' a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for 100''l.'' a-side. Dooley had a big reputation, but Slavin won in nine rounds. The subject of these remarks then had a little difficulty in finding an opponent, but eventually Mike Costello, an American, a good man, made his appearance, and the pair were matched for 100''l.'' a-side. They had a big battle in Sydney, lasting two hours and a half, and a draw was the result. Another match was made for 400''l.'', and 50''l.'' a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for 24''l.'' a-side to stop Costello in six rounds, but the latter is a very clever and tricky man, and lasted the time. Subsequently Frank journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for 100''l.'' a=side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for 250''l'' a-side and the Championship. Jackson's party wanted a match with gloves for a purse. They could not come to terms, though Slavin's backers offered Jackson 50''l.'' to sign articles for 500''l.'' Australian was now a bit bare, so in January last year the Australian made a trip to New Zealand. The first man put up was Snath, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the Champion. Terms could not be agreed upon until they had &quot;fiddled&quot; Slavin for nearly six weeks. The pair were then matched for 200''l.'' a-side, and fought with 2oz gloves at [Wangazul?], Laing failing to come up at the end of the fifth round. After this Slavin went back to Melbourne, hoping to get on a match with Jackson, and was surprised to find he had gone to America. Following this he met Jack Burke, but after Slavin had had all the best of it the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for 200''l.'' with small gloves, which ended in Mike being knocked out in 2min 40sec. He was very bad, and frightened all by not coming to for nearly half an hour. Slavin returned to Melbourne, and Burke having put several challenges in the paper, our visitor offered to stop him in six rounds; 200''l.'' a-side was put up and articles signed for 4oz gloves, but on the night Burke's party threw down regular pillow-cases. Of course this did not suit the other side, and there was a fine old wrangle, lasting two hours and a half. Eventually Mr Joe Thompson, who was referee, declared all bets off, and the two fought 6oz gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for 200''l.'', with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. This is considered to be one of Slavin's best performances. About this time Woolf Bendoff, of Whitechapel, was in Melbourne, and finding plenty of backers, they put up a forfeit of 100''l.'' for the Londoner to box Slavin, for 500''l.'' a-side. After seeing the latter polish off Burke, however, Bendoff's friends would go no further with the atch, and Slavin accordingly pocketed the &quot;century.&quot; An easy victory over Bill Buck in two rounds followed, and then Slavin, on October 17 made his first appearance in England at Astley's Theatre, where he beat &quot;Chesterfield&quot; Goode in five rounds in a match of twelve rounds, for 400''l.''  [4, Col. 6c] &lt;/blockquote&gt;

==== THE PELICAN CLUB AND THE FIGHT. ====
&lt;blockquote&gt;To the Editor of the Sportsman. 

Sir, — As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it. — Yours truly,  

Cecil Raleigh, Hon. Sec. 

Pelican Club, 21, Denman-street, W., Dec. 23. 

__________ 

We hear on good authority that it is the intention of the members of the Pelican Club to present Slavin, the Australian Champion, with a belt, in recognition of the manly exhibition he gave yesterday at Bruges. &lt;/blockquote&gt;

==== THE CHAMPIONSHIP OF THE WORLD. ====
&lt;blockquote&gt;Frank Slavin to John L. Sullivan. 

Frank Slavin, the Champion of Australia, is prepared to meet John L. Sullivan, under London Prize Ring Rules, for 500''l.'' or 1000''l.'' a-side and the Championship of the World. Slavin has lodged with us a sum of 100''l.'', and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr R. K. Fox, of New York, and the money lodged with him or the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; will lead to a match.&lt;ref name=&quot;:1&quot; /&gt; &lt;/blockquote&gt;

=== Dundee ''Evening Telegraph'' ===
This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:&lt;blockquote&gt;There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.&lt;ref&gt;&quot;London Gives the Colonist a Great Reception.&quot; ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&amp;page=3&amp;article=052&amp;stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.&lt;blockquote&gt;The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; 

Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot;Police!&quot; Smith was hurried from the ring; but Slavin and his friends refused to budge, and, after &quot;Time&quot; had been called, they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to Margate. Smith and his friends were left behind at Bruges.

Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec.

Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other.

Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.)

Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him.

Round 5. — A light round, Smith going down before a light blow. Time, 11sec.

Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead.

Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners.

Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground.

Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min.

Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together.

Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander.

Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec.

Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin.

Round 14 and last. — Just a second before &quot;Time&quot; was called, Slavin, remarking on the treatment he had received, said, &quot;Are you Englishmen, boys?&quot; This was received with derisive shouts from Smith's party, and then, in an instant, a cry of &quot;Police&quot; was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, &quot;Why don't you give me fair play?&quot; Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, &quot;I shall refuse to act any longer. This man&quot; — pointing  to Slavin — &quot;can't have fair play. I declare it a draw.&quot; Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there.

Here the fight came to an end, the referee exclaiming, &quot;I shall refuse to act any longer. I shall declare it a draw.&quot; There was considerable surprise among Slavin and his supporters, and the Australian said, &quot;I came 16,000 miles to win, and why don't you give a man a chance?&quot; After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine.

Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— &quot;As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it.&quot; 

Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. 

That Slavin has now a right to the title (empty though it be) of &quot;champion&quot; is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— &quot;Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. 

Arrival of Slavin at Margate. 

Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, &quot;See the conquering hero comes.&quot; Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, &quot;Well, Slavin, how have you fared?&quot; &quot;Very badly,&quot; responded the Australian. &quot;As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders.&quot; — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. 

The Pelican Club and the Fight. 

A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— &quot;In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business.&quot; 

It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. 

The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— 

&quot;In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, &quot;F. P. SLAVIN. 

&quot;Nayland Rock Hotel, Margate.&quot; 

Mr. Fleming's Statement. 

Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. 

The Preliminary Arrangements. 

The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the &quot;lost art&quot; was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the &quot;old style.&quot; Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. 

There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. 

Performances of the Men. 

Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. 

Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; hips, 41in.; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; calf, 16&lt;sup&gt;3&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in.; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton (&quot;eighty-one tonner&quot;) and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the &quot;eighty-one-tonner&quot; as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.&lt;ref&gt;&quot;The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring.&quot; ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&amp;page=1&amp;article=006&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== The ''People'' ===

On Sunday, 29 December 1899, the ''People''&lt;blockquote&gt;Disgraceful Scenes. Expressions of indignation.

The prize fight between Jem Smith, who poses as the fistic &quot;champion of England,&quot; and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a &quot;strong corner.&quot; The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that &quot;Smith did not lose&quot; proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper.

The Fight.

The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot; Police!&quot; Smith was hurried from the ring, but Slavin and his friends refused to budge, and after &quot;Time&quot; had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of &quot;Foul,&quot; the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate.

Slavin's Opinion of his Treatment.

The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there.

Indignation in London.

Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee.

The Pelican Club Meeting.

At the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of &quot;the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.[&quot;] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed.

Slavin Challenges J. L. Sullivan.

Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&amp;page=16&amp;article=150&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Sportsman'' ===
On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the &quot;testimonial&quot; fund for Slavin as well as some letters to the editor about the Slavin-Smith match:&lt;blockquote&gt;
A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). 

The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris.

The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.'']

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Lord James Douglas  2 2 0
* Hon. Michael Sandys  2 2 0
* Fred Kotchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Proprietors of The Sportsman  2 2 0
* Mr. J. Purdey  2 2 0
* Mr. E. W. Bayly  2 2 0
* Messrs H. A. Domby and Son  2 2 0
* &quot;Caption Coe&quot;  2 2 0
* Mr C. W. Bulpett  2 2 0
* Mr P. W. Carr  2 2 0
* Mr W. Morley Pegge  2 2 0
* W. D. S.  2 2 0
* J. E. S.  2 2 0
* Mr W. H. Grenfel  2 2 0
* Mr Quartermaine East  2 2 0
* Mr H. H. Hungerford  2 2 0
* Mr Noel Fenwick  2 2 0
* Sir John Leslie, Bart.  2 2 0
* Mr T. Brinkman  2 0 0
* Mr Calder  1 1 0
* Mr H. W. Hill  1 1 0
* Mr John Young  1 1 0
* Mr D. A. Neilson  1 1 0
* Mr W. Neilson  1 1 0
* Mr Julius Simon  1 1 0
* Mr J. Edge  1 1 0
* Mr H. Marshall  1 1 0
* Mr F. Aynsley  1 1 0
* Col. Hon. Wellesley  1 1 0
* Mr Charles Hawtrey  1 1 0
* Mr W. F. Ashton  1 1 0
* Mr Barratt  1 1 0
* Old Durham Ox  1 0 0
* Sir John Willoughby  1 0 0
* A friend, per C. H. A.  0 10 6
* Mr Ben Giles  0 10 6
* Mr Tom Vale  0 10 6
* [E.?] W.  0 10 6
* Officer  0 10 0
* Per Mr Ike White, Exchange Club:
** S. W.  2 2 0
** Mr G. B. Knight  1 1 0
** Mr A. Powles  1 1 0
** Mr S. Blandford  1 1 0
** T. F. W. C.  1 1 0
** Mr Leo Harward  1 1 0
** Mr J. Harward  1 1 0
** Mr H. Harward  1 1 0
** Mr [Monteagle?] [Moutcagie??]  1 1 0
** Mr C. Orate  0 10 6
** W. G. P.  0 10 6
** H. R. T  0 10 6
** Mr Josiah Smith  0 10 6
** W. H. R.  0 10 6
** Mr R. Wilson  0 5 0
* Per Mr M. J. Harlow, Margate:
** Mr A. H. [Cosi? Coni?]  2 2 0
** Mr J. Thurston  1 1 0
** Mr J. A. Fagg  1 1 0
** Mr F. [Miles?]  1 1 0
** Mr Adolphe [Drineqbeir?]  1 1 0
** Mr Ernest [Drineqbier?]  1 1 0
** Mr Lacy Brown  1 1 0
** Mr Algy Brown  1 1 0
** Mr M. J. Harlow  1 1 0
** An Admirer  1 1 0
** Mr J. W. Robinson  1 1 0
** Mr W. Searles  1 1 0
** Mr J. Smith  1 1 0
** Mr J. Hume  1 1 0
** Mr J. Painter  1 1 0
** A Scotchman  1 1 0
** Mr J. Mundell  0 10 6
** Mr Bob Redman  0 10 6
** Mr G. A. Body  0 10 6
** Mr J. Phipps  0 10 6
** Mr J. Doughty  0 10 6
** Old Fisherman  0 10 6
** Mr Joe Knott  0 10 6
* Per Mr B. J. Angle
** Some mebers of the Stock Exchange  125 0 0
** Hr Henry J. Turner  2 2 0
* Per Mr W. J. Innes:
** Mr Sidney Fry  2 2 0
** Mr W. J. I.  2 2 0
** Mr W. Schlesinger  2 2 0
** Mr Joseph Thompson  2 2 0
** Mr Barney Thompson  2 2 0
** Mr L. Meyers  2 2 0
** Mr W. Shoe  2 2 0
** Mr George Cooper  2 2 0
** Mr J. T. Crossley  2 2 0
** Mr Andy Anderson  2 2 0
** Mr G. M. T.  2 2 0
** Mr F. R.  2 2 0
** Mr W. Bertram  1 1 0
** Mr J. W. Hammon  1 1 0
** Mr Alec Harris  1 1 0
** Mr John Isaacs  1 1 0
** Mr Edward [Dance? Dasce?]  1 1 0
** Mr George Blanchett  1 1 0
** Mr F. [Leleu? Lolou?]  1 1 0
** Mr W. J. R.  1 1 0
** Mr H. C.  1 1 0
** Mr A. A. M.  1 1 0
** Mr D. W. Fowler  0 10 6
* Per Mr F. H. Gale:
** Mr R. Moore  2 2 0
** Mr A. H. Goode  2 2 0
** Mr [B.? E.? R.?] J. G. Goode  2 2 0
** Mr F. H. Gale  2 2 0
** Mr H. Ulph  1 1 0
** Mr J. Hooper  1 1 0
** Mr C. Kittall  1 1 0
** Mr C. Moore  1 1 0
* Per Mr W. Riley:
** Mr Brand (Wiltshire)  2 2 0
** Mr Charles Godfrey  2 2 0
** Mrs Lane (Britannia)  2 2 0
** Mr Sam Adams  2 2 0
** B. and M.  0 7 6
* Per Mr A. Cockburn:
** Mr Richardson  2 2 0
** Mr A. Cox  1 1 0
** Mr Hy. Brackenbury  1 1 0

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and &quot;knuckle-dustered&quot; him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, &quot;Come and have a round.&quot; The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called &quot;seconds&quot; should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of &quot;Police&quot; was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska &quot;tough,&quot; as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how &quot;seconds&quot; — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a &quot;duster&quot; might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &amp;c.,

A Lover of Boxing.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &amp;c.,

Leicester, December 28. J. Barrett.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded.

I cannot, however, agree with those who hope &quot;that we have seen the last of prize-fighting,&quot; and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &amp;c.,

J. W. B. C., Junior Carlton Club, December 27.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself &quot;England's Champion.&quot; I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon &quot;British fair-play and courage.&quot; — Yours, a lover of it, Sydney Galvayne. 

York, December 26. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, 

Birmingham, December 27. Tom B. Sharp. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &amp;c., 

I. White.  

Exchange Club, Queen-street, Cheapside, Dec. 28.&lt;ref&gt;&quot;The Ring. The Slavin Testimonial.&quot; ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&amp;page=4&amp;article=020&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; &lt;/blockquote&gt;
On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:&lt;blockquote&gt;JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.
{{paragraph break}}

J&lt;small&gt;OHN&lt;/small&gt; L S&lt;small&gt;ULLIVAN&lt;/small&gt; has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me.&quot;&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:&lt;blockquote&gt;THE SLAVIN TESTIMONIAL
{{paragraph break}}
The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office.&lt;/blockquote&gt;
{| class=&quot;wikitable&quot;
!
!£
!s.
!d.
|-
|Amount already acknowledged
|407
|3
|6
|-
|Mr E.Tabernacle
|1
|1
|1
|-
|H. S.
|1
|1
|0
|}
&lt;blockquote&gt;The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Huddersfield Daily Chronicle'' ===
On 10 January 1890:&lt;blockquote&gt;
Mitchell and Lord Mandeville.

As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of &quot;sports&quot; at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by &quot;Pony&quot; Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &amp;c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. &quot; [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and &quot;went for&quot; Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and &quot;Pony&quot; Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was &quot;something up,&quot; and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&amp;page=4&amp;article=071&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;
After the Pelican Club meetings, Slavin comments:&lt;blockquote&gt;On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English &quot;champion,&quot; made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.&lt;ref&gt;&quot;Slavin at the Royal Acquarium.&quot; ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&amp;page=3&amp;article=042&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:&lt;blockquote&gt;Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. &quot;This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of &quot;heelers&quot; always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me.&quot;&lt;ref&gt;&quot;The Recent Prize Fight.&quot; ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&amp;page=6&amp;article=053&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Cork Constitution'' ===
About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:&lt;blockquote&gt;This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s &quot;lefts.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Colonial Standard and Jamaica Despatch'' ===
This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:&lt;blockquote&gt;The disgraceful affair, called by courtesy and [sic] &quot;International Prize Fight,&quot; which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The &quot;fight&quot; lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn [??, s/b Royston?], Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.&lt;ref&gt;&quot;London.&quot; ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&amp;page=2&amp;article=019&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;
== Notes and Questions ==
# 

==References==
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International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium.

==Overview==
On 23 December 1889 a championship boxing match in Bruges, Belgium, ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin or getting him near the ropes and beat him up, using fists, sticks, &quot;knuckle-dusters,&quot; &quot;life preservers&quot; and boots. The referee ruled the match a draw to prevent Slavin from being killed and then fled. The ''Cork Constitution'' called it &quot;the Smith-Slavin piece of brutality.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;

A knuckle-duster was &quot;A metal instrument made to cover the knuckles,&quot;&lt;ref&gt;“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.&lt;/ref&gt; like brass knuckles, and a life preserver was &quot;A stick or truncheon weighted at one end&quot; with lead.&lt;ref&gt;“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.&lt;/ref&gt;

One meeting of the boxing-oriented London Pelican Club was held almost immediately. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below).

Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England.

Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made here to offer complete coverage. Specialized sports coverage is privileged.

==Logistics==
* 23 December 1889, Monday, 8:00–9:00 a.m., actual fighting commencing at 8:35
* &quot;the private grounds of a gentleman's residence near Bruges, Belgium&quot;&lt;ref name=&quot;:1&quot;&gt;&quot;The Great Fight. Smith &amp; Slavin for £1000. Disorderly Proceedings. Brutal Attack on Slavin. Smith Knocked Down Repeatedly. He Leaves the Ring. A Draw Declared after 14 Rounds.&quot; ''The Sportsman'' 24 December 1889, Tuesday: 3 [of 4], Col. 1a – 4, Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&amp;page=3&amp;article=057&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; (3, Col. 1a)
* Stakes: £1,000
==Related Events==
* [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]]
* the &quot;[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]&quot; 
==Who Was Present==
# Jem Smith, British boxer
## Jack Harper, Smith's second
## Jack Baldock, Smith's second
## Mr J. Fleming, umpired for Smith
# Frank Slavin, Australian boxer
## Josh Cosnett, Slavin's second
## Bob Kirby, Slavin's second
## Mick Taylor, &quot;superintended the arrangements in his [Slavin's] corner&quot;
## Mr W. Riley umpired for Slavin
# &quot;the company round the ring
## Lord Mandeville — [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]]
## Lord Royston
## Hon. M. Sandys
## Mr Abington — Mr. G. Abington Baird, Smith's manager? and horseman (see description in Anthology)
## Mr F. Kotchie
## Mr J. Kilby — Jack Kilby, &quot;a real sportsman&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## Mr C. Parsons
## Mr Ike White
## Mr C. Morton
## Mr Harris
## Mr E. Bayley — Teddy Bayley
## Mr John Lewis (Australia) — Jack Lewis, Slavin's manager
## Mr J. Vesey — Joe Vesey, more often, or Vezey, referee
## Mr F. Hinde — Frank Hinde
## Mr M. Taylor
## Mr J. Calder — timekeeper
## &quot;Pop&quot; Joy
## R. Habbijam — Bob Habbijam
## G. (&quot;Barney&quot;) Sheppard — &quot;superintended the pitching of the ring&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3b)
## Jem Carney
## Tom Smith
## Mangie [?] Hyams
## Matthews
## Summers
## Boulton
## Mr G. T. Dunning — Tom Dunning, worked at ''The Sportsman'' and served as treasurer for the subscription to collect a cash prize for Slavin.
## &quot;several others, including two or three reporters&quot; — &quot;three scribes with pockets bulging from a surplusage of note-books dropped in&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## The writer of the big 24 December article, who reports for ''The Sportsman'' and who is not the &quot;well-known sporting journalist&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1b) who had heard that Slavin had been arrested in Margate
# &quot;an English resident in Bruges&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3a)
# &quot;Pop&quot; Hinde
# C. Horton
# &quot;a mob of Birmingham men&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
# Lord Rosslyn ?
# &quot;The Squire&quot;
#
==Entertainment==

==Anthology==

=== The Sportsman ===
The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 and most of page 4 (nearly 13 full columns) entirely to this story. This reporter's writing style is very ornate:

* &quot;Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping&quot; (3, Col. 1c)
* &quot;For a time it [the sea] was a bit rough, and one of the good sailors commenced quoting Byron. 'O'er the glad waters of the dark blue sea,' sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2a)
* &quot;Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2b)
* &quot;While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
* &quot;He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)

Besides the street-fighting terms (knuckle-dusters, life preservers) above, he also uses slang and local vocabulary like sabots (wooden clogs worn by working-class Belgians), Brums (people from Birmingham [?])

This is, then, a very long article, reproduced here in full except where the text is illegible, the very large block quote broken up by level-2 heads for navigation:&lt;blockquote&gt;The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below.&lt;/blockquote&gt;

==== THE JOURNEY TO THE CONTINENT. ====
&lt;blockquote&gt;(By our special reporter.)

[?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of &quot;globe-trotting,&quot; as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came]

A Rude Shock.

[Arriving??] at the office of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that &quot;something had happened.&quot; His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. &quot;What do you think of this?&quot; asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. &quot;Stoppage, but think can get bail. — Slavin.&quot; Here was a pretty &quot;go,&quot; the mystic words being translated meaning that the Australian had been &quot;pinched&quot; at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it

A May Morning.

Long before '''the earliest purveyor of the lacteal fluid''' arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of &quot;London peculiar&quot; had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking &quot;all broke up,&quot; put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a '''surplusage of note-books''' dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, &quot;Pop,&quot; and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey [sic], who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross &quot;The Squire,&quot; Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''.

Down to Dover.

It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly &quot;gave the game away,&quot; and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as &quot;dirty&quot; weather in store.

On the Briny.

The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and &quot;once on board the lugger&quot; my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the &quot;Princess&quot; swung off from the pier the two &quot;Bobs,&quot; Kirby and Habbijam, betook themselves to the &quot;rail&quot; in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. &quot;O'er the glad waters of the dark blue sea,&quot; sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having &quot;sought the seclusion which the cabin grants,&quot; and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in &quot;Pop,&quot; and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. [3, Col. 2a–2b] 

Ostend to Bruges.

While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded &quot;foreigner&quot; who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, &quot;Well, when is the 'mill' coming off?&quot; In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any &quot;mills.&quot; As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring '''that useful saponaceous compound''' with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity. [3, Col. 2b–2c]

Pugilism at the Piano.

While '''awaiting the refection''' Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The &quot;pelting of the pitiless storm&quot; continued the greater part of the night. 

Arrival of Jem Smith. 

The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arrangements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or &quot;the manger&quot; as &quot;Pop&quot; expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery &quot;Good morning, gentlemen,&quot; Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. 

The March to the Battlefield. 

Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out again. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work [3, Col. 3a–3b] 

Building the Ring. 

Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. 

The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, &quot;Pop&quot; Joy, R. Habbijam, G. (&quot;Barney&quot;) Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that 

The Dreaded Birmingham Division 

had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the &quot;boys&quot; worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, &quot;I hope all those present will keep quiet and help me all you can,&quot; at which brief speech there was a chorus of &quot;hear, hear,&quot; especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. 

The Call of &quot;Time&quot; 

At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, 

Smith was Knocked Down 

By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their &quot;Champion&quot; did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the &quot;Champion&quot; wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to &quot;be a man.&quot; Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of &quot;Here they are&quot; was raised, and 

A Stampede Ensued 

from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, &quot;There they are, there they are.&quot; By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, &quot;Are you Englishmen? I am an Englishman, and only want fair play.&quot; The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a &quot;duster,&quot; the referee saying, &quot;I saw that.&quot; As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, &quot;I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play.&quot; The referee concluded by giving the decision, &quot;It's a draw.&quot; The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting &quot;There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance.&quot; There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. 

There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended 

Making All Speed Homewards. 

The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. 

Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. &lt;/blockquote&gt;

==== THE FIGHT. ====
&lt;blockquote&gt;First Round. 

Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. 

Second Round. 

No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, &quot;First blood,&quot; a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On &quot;Time&quot; being called the pair were quick to scratch, that is to say, Slavin walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec.

Third Round.

Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking back to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec.

Fourth Round.

Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, 1min 15sec.

Fifth Round.

&quot;Don't go into his corner,&quot; urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite corner, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds.

Sixth Round.

Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec.

Seventh Round.

In vain did Slavin's friends implore him not to go into Smith's very &quot;warm corner.&quot; It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion was something terrible, and the ring looked very much like being invaded. The ropes in Smith's corner were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. 

Eighth Round.

Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth. Time of round, 1min 30sec.

Ninth Round.

This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a]

Tenth Round.

The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. 

Eleventh Round. 

Neither was quite so quick to time, and Harper, noticing that, said, &quot;This is not a two-round job, Patsey.&quot; Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to &quot;Come on and have a round.&quot; This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. 

Twelfth Round. 

A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. 

Thirteenth Round. 

Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, &quot;Oh!&quot; Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of &quot;Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a &quot;blue.&quot; The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words &quot;fight on&quot; came he and his friends were very delighted. 

Fourteenth and Last Round. 

When &quot;Time&quot; was called Slavin folded his arms and waited for Smith to &quot;come on,&quot; but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. 

No sooner was this round over than the referee called out. &quot;I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw.&quot; Then, going up to Slavin he said, &quot;I will not stay to see you killed.&quot; The Antipodean stood up and exclaimed, &quot;If you are Englishmen why don't you give me fair play? I have not had fair play.&quot; 

After this came a stampede. 

Remarks. 

Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as &quot;fights in the old style.&quot; That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the &quot;upper crust.&quot; Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being &quot;bashed&quot; on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with &quot;dusters&quot; and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a &quot;glover.&quot; With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a] that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or &quot;bottom&quot; of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, &quot;he was just getting warm.&quot; Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect.  

For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved.

From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play.

Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow. Bearing on this subject, Rule 26 says:— &quot;That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle.&quot; Rule 24 is also plain enough in all conscience, &quot;That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the &quot;boys&quot; and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but &quot;Kim&quot; soon showed he could hold his own with the &quot;boy,&quot; who beat a hasty retreat at the &quot;sight&quot; of six-inch &quot;Jack.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 1a)

We now take our leave of this unsavoury affair, which will probably be the last of its kind — &quot;a Championship knuckle fight.&quot; The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner &quot;merry mills in the old style&quot; die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office about three o'clock.

Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of &quot;dusters&quot; the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye.

It may be stated that not a single bet was made at the ring side. [4, Col. 1b]&lt;/blockquote&gt;

==== SLAVIN'S OPINION OF HIS TREATENT. ====
&lt;blockquote&gt;His Future Plans. 

On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. 

Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. 

With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. 

Much sympathy was expressed on all sides, and in the course of conversation with a S&lt;small&gt;PORTSMAN&lt;/small&gt; representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] 

Did you expect any trouble at the ring side? 

Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. 

Do you feel any effects of the fighting? 

Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. 

Why did you persist in going to his corner? 

Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. 

Of course you heard a mob was got together for Smith? 

Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. 

What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. [4, Col. 1c] 

I suppose you have a poor opinion of English fair play? 

Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. 

What are your plans for the future? 

Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. 

With the gloves? 

Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. 

You prefer the old style to the new? 

Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. 

Shall you give any exhibitions? 

Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. 

You put on weight the last week? 

Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. 

What do you think of our seconds? 

Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. 

When do you propose visiting London? 

Slavin: I'll come up to-morrow and call at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office about three. 

A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. &lt;/blockquote&gt;

==== A Testimonial for Slavin. ====
&lt;blockquote&gt;So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.'' The maximum donation was 2 guineas, which would be 2£ 2''s.''] 

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Hon. Michael Sandys  2 2 0
* Mr Fred Ketchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Mr J. Calder  2 2 0

Any contributions sent to the hon. treasurer will be duly acknowledged in T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;, which in a couple of days will contain the names of a committee now in course of formation. 

In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; 

Sir, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  &quot;Champion of England&quot; to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. 

If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C. 

Royal Thames Yacht Club, December 23. 

[We refer our correspondent to the paragraph above. — Ed. T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. ] [sic bracketed interpolation]  [4, Col. 2b] &lt;/blockquote&gt;

==== Eyewitnesses Interviewed. ====
&lt;blockquote&gt;A representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known &quot;sports&quot; (including more then one scion of a noble house) for a few minutes' chat. 

Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which did not number more than forty, being particularly noticeable. 

&quot;How did the combatants' seconds behave?&quot; — &quot;Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes.&quot; 

&quot;What form was the Englishman displaying?&quot; — &quot;From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds. Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy.&quot; 

&quot;How do you account for such a noisy corner at the back of Smith?&quot; — &quot;Although they did not number more than eight, they were some of the worst characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, inasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, &quot;framed&quot; [sic double quotes] up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his party.&quot; 

Are these men known at all?&quot; — &quot;Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period.&quot; 

&quot;Was not this sort of trouble expected?&quot; — &quot;It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated.&quot; 

In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. 

&quot;How did the crowd behave towards the finish?&quot; — &quot;There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious.&quot; 

&quot;Is it true that attempts were made to tamper with the water used by Slavin's seconds?&quot; — &quot;Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail.&quot; 

&quot;Did Smith deal out much punishment?&quot; — &quot;His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought for two hours more. He could have beaten Smith and both his seconds into the bargain, given fair play.&quot; 

&quot;Was Slavin injured during the fight?&quot; — &quot;No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed.&quot; 

&quot;As to the referee's decision?&quot; — &quot;That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened.&quot; 

&quot;What becamiof Smith after the fight.&quot; — &quot;He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does.&quot; 

&quot;How did the Cornstalk behave in the ring?&quot; — &quot;He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion.&quot; 

Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a &quot;mill,&quot; was very severe in his comments on the gang who were so lost to all sense of decency and fair play. 

Mr White said: — &quot;The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He had to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful.&quot;  [4, Col. 2c] 

&quot;Do you think it was for the best that the contest was stopped as it was?&quot; 

&quot;Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]&quot; 

&quot;What do you think of Slavin's abilities?&quot; 

&quot;He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt.&quot; 

On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. &quot;He is coming to town to-morrow,&quot; said the speaker, &quot;and shall have a great reception. Some good sportsmen have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it.&quot; 

&quot;What seemed the general opinion on the ground?&quot; 

&quot;Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took.&quot; &lt;/blockquote&gt;

==== Reception of the News in Birmingham. ====
&lt;blockquote&gt;Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a] and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. 

How the Match Was Made 

On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— &quot;He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, was unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers.&quot; [sic: quotation begins at the top of the paragraph] 

Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; and signed:   

Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', for the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally.   

In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names. [4, Col. 3c] &lt;br&gt;
Frank Hinde (for Frank Slavin). &lt;br&gt;
John Fleming (for James Smith). &lt;br&gt;
Witness for Smith, F. Montague. &lt;br&gt;
Witness for Stavin, John Lewes. &lt;p&gt;

Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The &quot;Cornstalk,&quot; though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not step in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds.   

Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell.   

Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[,] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice.
&lt;/blockquote&gt;

==== The Men in Training ====
&lt;blockquote&gt;Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp.   

Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound,  Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson.   &lt;/blockquote&gt;

==== Finding the Battle-Ground. ====
&lt;blockquote&gt;(By one of the Explorers.)   

At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class &quot;smoker&quot; attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed that a &quot;choppy sea&quot; was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. &quot;How confoundedly annoying,&quot;  he said. &quot;This is the sixth time I have been across during the year, and not once has it been calm.&quot; &quot;Better have a sleeping berth, sir,&quot; said a sprightly youth at the top of the stairs. &quot;Sleeping berth be hanged. I'll remain on deck,&quot; said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit &quot;peckish&quot; and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the &quot;magging&quot; [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, &quot;struck ile [?]&quot; with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The &quot;elderly party&quot; had a voice greatly resembling the dividing of tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. Just as I thought of giving up sleep for a bad job a gale sprang up. Though the Geselle is a record breaker she soon felt the effects of the breeze and began to roll heavily. The increase in the motion of the steamer was followed by a decrease in the conversation around me, and by degrees the punster went from had to worse. The &quot;[hussey??]&quot; [one?] also dropped into a walk, and after monosyllables had been exchanged the two came to a dead stop. Offering up a prayer of thanksgiving for this I closed my eyes again, and being a good sailor was quickly sleeping the peace of, I hope, the just. The concussion of the Gazelle against the Guernsey Pier, accompanied by a prod from Fleming's stick, at length aroused me.  [4, Col. 4b]   

Among the Channel Islands.   

The time by &quot;Joe Vickers&quot; was 7 a.m., and half-awake I took to my boots. &quot;Need not lace them; we shall be there in a minute,&quot; said Fleming, and in a drowsy, submissive, take-me-anywhere-if-only-to-bed sort of way I followed him on deck. A Babel of voices and a clutch at my bag and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a &quot;[sheenie?]&quot; (goodness knows what he meant, though), made us welcome, and agitating the communicator he handed us over to the care of a dark-eyed chambermaid. I forget her name, and as Fleming is not here I cannot give it. I will say no more than that after a cup of tea diluted with delicious cream, we took a well-earned and mucch needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the &quot;holiday maker&quot; and the &quot;honeymooner,&quot; is, as everybody knows, or should know, situate in the English Channel near the coasts of Brittany and Normandy. It is 68 [?] miles from Weymouth, 7 from Sark, 19 from Alderney [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in &quot;Mike.&quot; Said I, &quot;He's a Frenchman.&quot; &quot;Get out,&quot; replied Fleming, &quot;he's a 'bark.'&quot; &quot;Drinks he isn't,&quot; &quot;Done; drinks he is.&quot; &quot;Mike&quot; had not at that time spoken, but had noiselessly passed to and fro with a military step and a hand behind him. In reply to a call from Fleming he approached, and a &quot;Yes, yer honour,&quot; told me I should have to &quot;part.&quot; &quot;Mike&quot; was a trump, and I loved him from the start. For eleven years he had been in the service of a colonel at 1''l.'' a week, and three suit of clothes in the year. Hotel pay was not so good, and though he made &quot;a bit&quot; in the summer, the young ones were generally there when the bills were paid, if not when the work was on. &quot;Mike&quot; told us lots of things in what many consider the &quot;fascinating brogue.&quot; He informed us — and how our hearts beat at the news — that the island, which had a population of over 32,000, employed but six policemen. &quot;There's not a beggar in the land,&quot; said Mike, &quot;and but one Hansom cab, and,&quot; pointing to the window, &quot;be jabers, there it goes. &quot; So far so good, but after a walk round the son of Erin gave us a shock by saying, in reply to a query from Fleming, &quot;They've a way of their own on this island, and nearly ivery [sic] other man is a special constable.&quot; Fortunately we had just finished our repast, and, supporting each other, we tottered faintly to the door, partially regaining strength with a breath of the pure sea breeze. Beads of perspiration stood on the brow of the Pelican, whilst I was suffering from a &quot;trickling&quot; down the back of anything but a pleasing nature. Visions of a dungeon-cell under Castle Coronet flashed through my brain as a voice and a step were heard behind, and then I was somewhat reassured. &quot;When would you like a chair?&quot; fell softly on mine ear. Afraid that we had been thought of as a bit week on the leg, I turned to the speaker and asked what he meant. Explanations ensuing, we found the &quot;chair&quot; was a sort of Bath chair on wheels used throughout the island. Declining the offer, we took a stroll, and putting this and that together, came to the conclusion that our task was of a very delicate nature. The first thing was to interview the captain, who, in reply to a note, courteously waited upon us. He gave us the geography of the island and an introduction to Mr M., a friend, who we unfortunately found away from home. Taking a &quot;chair&quot; we then started on a voyage of exploration, making, as advised, our way to L'Ancresse [?] Common. Leaving St. Peter's Port and passing through St. Sampson's — the Vale — we came across acres of vineyards, in one instance extending nearly a mile along the road-side. Half an hour's drive brought us to the common mentioned, and dismounting we tried in vain to find a level piece of turf. Windmills abounded on all sides, and plodding on we reached L'Ancresse Bay, a lovely little spot, where the sea rippled softly over a bed of almost snow-white sand. Nowhere could be found a suitable spot whereon to pitch the &quot;magic square,&quot; and as the shades of night were falling fast we rattled back to the &quot;Royal.&quot; A wet night offered no inducement for further travel, and it was a case of early to bed and early to rise, the next day finding Fleming and myself at the door of Mr M. Acquainting this gentleman with the Nature of the business in hand he, like a good sportsman, fell into the scheme and at once volunteered to drive us to &quot;the best spot in the world.&quot; &quot;Bessie&quot; was harnessed to the village cart, and away we went at the rate of fully fourteen miles an hour in the direction of the Forest. I have sat behind many a good goer, but must for her size award the palm to Bessie a beauty barely twelve hands high. Throwing the reins in the bottom of the vehicle, Mr M. let the pony have her head, and the way she went uphill, down dale, and past other traps was truly astonishing. Great St. Peter's was passed, and lovely scenery encountered, the gardens being full of flowers, geraniums and roses blooming on all sides. An enjoyable drive of forty-five minutes found us at Boequaire [?] Bay, and directly in front stood the island of Lihou. As many of our readers may know, Lihou is a small hill of land that at low water can be reached on foot from Guernsey. As a fact, a paved roadway is laid in the bed of the ocean, across which both foot passengers and carts easily make their way. With the tide up communication is effected by boats, but this being low at the time we had the novel experience of   

Driving Through the Sea.   

Gaining the other side we &quot;explored&quot; and as the piece of land is only some three hundred yards by a hundred and fifty, we were not long about it. Two excellent patches of level turf were found, but just as we were making for the trap a gaunt, Robinson Crusoe-looking individual scaled one of the small stone walls and barred the way. Of course, he wanted to know our business, and, after a rather rough opening, quickly &quot;listened to reason.&quot; In the course of conversation we found that the population of the island was limited to three, the old &quot;josser&quot; and his two daughters, with a &quot;moke&quot; thrown in. Pointing a long skinny finger, which might have belonged to the Witch of Endor, towards the quadruped mentioned, the old man said, &quot;Us two keeps the family. He draws the seaweed and I burn it.&quot;  &quot;And why do you burn it?&quot; &quot;Well, to make [Col. 4c–5a] ashes, and from the ashes iodine is manufactured.&quot; &quot;And how much do yen earn a week by that?&quot; said  I. &quot;Oh, about 4''s.''; not 5''s.'' any week.&quot; Great Scott! and this was all the old man, his two daughters,  and the &quot;moke&quot; had to live on. Handing him a franc each, which nearly stupefied the old chappie, Fleming and myself returned to Mr M., who drove us rapidly back to St. Peter's Port. The next day we crossed to Sark and Jersey, but found nothing so suitable as the ground at Lihou, and fairly well satisfied with the result of our visit, a day later boarded the Lynx for the return journey.   

During our visit the weather had been of a fairly favourable description, but the voyage home was made under very depressing circumstances. Rain fell heavily and a thick mist hung over the water, which after going a few miles was alive with porpoise. Some of these animals came within a few yards of the Lynx, but the speed of the boat proved too good for the finney ones, who were eventually left behind. With a breeze springing up sail was hoisted, and dashing through a heavy sea in splendid form the Lynx made some sixteen knots in the hour. In this way half the journey was completed, when all of a sudden we were caught in a squall. In an instant the wind chopped round, and before you could say &quot;Jack Robinson&quot; the good vessel was turned broadside to. The order to take in sail was promptly obeyed, but the clatter on deck brought many an anxious passenger rapidly from below. A very heavy sea was now running, and though the Lynx kept her course bravely the decks were swept from stem to stern. A huge East India-man now hove in sight, and, as one of the men informed me, she had the audacity to cross our bows instead of passing under our stern. This breach of ocean etiquette ruffled the skipper of the Lynx, who chased the trader, and eventually made her alter her course. This little &quot;diversion&quot; caused much amusement amongst our passengers, and after a cold, rough passage we landed safely at Weymouth. Chilled to the bone, we made for the Victoria Hotel, where a chop and a glass of old October revived the almost stagnant circulation. London was reached about 9.30, and, tired and worn out, I was glad to lay my head on a pillow I had not seen for four days.   

From London to Ostend.

Unforseen [sic] difficulties having cropped up, the Channel Island scheme had to be abandoned; consequently we were once more called upon to leave our native land. This time Belgium was our destination, and having carefully studied the map of that country, we took our departure one fine starlight evening from Charing-cross by the 8.15 Continental express. In the previous week we had the luxury of a compartment to ourselves, but on the occasion under notice we were not so fortunate. Two individuals, one with a strong Midland accent, and the other a Frenchman speaking bad English, shared the carriage with us, and at such a rate did the former's tongue go directly he entered that I gave up reading as a bad job, with a sigh put away my unfinished book of the previous trip. Fleming fared better, and having a knack of ignoring all around him, once more tackled &quot;Blood Money,&quot; and almost read himself off to sleep.  In the meantime our friends in the opposite corner made &quot;hay,&quot; and I soon found out they were both in the lace trade. With a tongue that went a couple of hundred to the minute, the Midlander almost monopolised the conversation, from which I gleaned the two were bound for Calais, and thanked goodness we should part at Dover.

Reaching this port about ten we boarded the Ostend boat, which, as regards passengers, was very poorly patronised. Hardly a breath of wind was stirring, and with the sea calm as a mill-pond we had a pleasant passage over, and landed safely at Ostend soon after 3 a.m. A cup of coffee was hastily swallowed, and a half-hour's run in the Brussels express found us  [4, Col. 5b]

At Bruges.

Leaving the train, we had considerable difficulty in finding our way out of the station. For a moment we caught sight of a drowsy old porter, but the last carriage had not passed away from the platform ere he was lost to view. Seeing an opening on other side we made for it, only to have our passage barred by a half-dressed Belgian who, whilst displaying a total ignorance of English, Irish, and Scotch, directed us to return the way we came. For some time we tried in vain to find an outlet, and just as I had made up my mind to walk the platform until daylight Fleming found an iron gate ajar. Without loss of time we were in the street, and crossing the round, knocked loudly at the door of the nearest hotel. All was in darkness, and with no response to our fourth appeal for admittance, a return to the station or a &quot;tramp round&quot; appeared to be our only alternative. A slight noise inside followed by a shuffling of feet, again raised hope within us, and when, after some trouble, the door swung back and brough to view a shock-headed Belgian our relief was great. No time, I can assure you, was lost in getting between the sheets, and rising betimes we again went on a voyage of discovery. Fleming had a chum in the town, and to his house of call our steps were directed. Again disappointment was in store for our wanderers, and a feeling of disappointment came over me when we found Mr G. was away. There was, under the circumstances, nothing left but to wait, and Fleming, who puts on hair rapidly, decided to place himself under the care of the town barber, or, I should, properly speaking, my ''coiffeur''. For this piece of indiscretion he had to come away minus a slice of his lower lip, which, fortunately, was not large enough to seriously reduce his avordupois [sic?], and heaping blessings (?) [sic] on mankind generally, not in any way forgetting the &quot;Bruges Barber, we roamed over the old town, noted for its magnificent cathedral and many seats of learning. The hours sped on pleasantly enough, and in due time our friend was unearthed. Being pledged to secrecy I am not going from my word by giving full particulars of

The Battle Ground

placed at our disposal. This is in close proximity to the town — in fact, not half an hour's walk, and on reaching the spot we were shown a lovely piece of turn. A clump of trees on each side partly hid the house from view, and when I say that the ring was to be pitched on a lawn tennis ground, it will be seen that neither side could claim an advantage as regards standing ground, &amp;c., the lawn being as level as a billiard table. Though our efforts so far had been crowned with success, we did not get permission to use the &quot;pictch&quot; without much trouble. The matter had to be seriously considered, and Mr G. would not give a decided answer without waiting another day. Consequently our stay had to be extended and the following morning, after repeated promises that the affair should be conducted in an orderly manner with not a rough at the ring-side, we got the answer, &quot;Yes.&quot; &quot;But,&quot; said Mr G., &quot;if there is any row or crowd I can assure you you will have a journey for nothing; therefore if trouble is at all probable keep away and save your money and time.&quot; As long as faith was not broken by either side we saw no reason why the matter should not be settled in a proper manner, and making all the necessary arrangements and laying down plans of a most complete description, we returned to town well satisfied with the result of a visit to Bruges.  [4, Col. 5c]

Trouble about the Referee. 

On returning to town we received satisfactory reports of the work done by each of the contestants. Slavin, looked after by Josh Cosnett, the Birmingham heavy-weight, and his friend Mr Michael Taylor, of Australia, though rather light, was in good health. At one time he scaled only 12st 8lb, but a judicious spell of light work brought him back to 13st odd. Considering the importance of the business in hand, and its bearing on future events, Slavin had some  cause for anxiety. In addition to this he was fighting for 300''l.'' of his own money, and consequently wished to do himself justice in the international battle. 

The next question was the selection of someone capable and willing to fill the all-important post of referee. According to the articles it was left to the stakeholder to appoint the official named providing the representatives failed to agree, and in the end this course had to be adopted, solely owing to the objections lodged by Smith. The first gentleman suggested by Slavin's side was Mr B. J. Angle, whom Smith promptly objected to. Next a well-known owner of racehorses (a gentleman who adjudicated upon the knuckle fight between George Camp and &quot;Click&quot; Soles) was proposed with no better result. Mr Bat Murphy, strange to say, would not suit, and then in succession &quot;No&quot; was said to the names of  Messrs G. H. Vise, A. F. Bettinson, and [B.?] Bayley. All these gentlemen, it must be stated, were proposed by Slavin and objected to by Smith, who in the meantime had named a member of the Pelican Club, a gentleman of limited experience in these matters, and the landlord of the house from which he was training, Mr W. Riley, Newmarket. This brought about a deadlock, and the state of affairs getting wind, ugly rumours were soon afloat, and even money, and in many cases good odds, were betted that there would be &quot;no fight.&quot; All this while Slavin was chafing and fuming at the delay, whilst at the same time his  representative and Smith's were at work almost night and day endeavouring to bring about a settlement of the difficulty. For a long time it looked as though the matter would end in smoke, as after the services of the stakeholder had been called upon great difficulty was experienced in getting anyone to accept the office of referee. So late as Friday at noon nothing had been done, and it was not until the evening of that day that Mr Joe Vesey, consenting to stand, was nominated by the stakeholder. 

A Scare.

Though the battle-ground was only known to two persons, the country where the fight was to be decided being, in fact, or supposed to be, a perfect secret, on Friday morning a paragraph said to have been sent out by a news agency appeared in several papers, and created intense astonishment and consternation. This, set out in large type by those who we should have thought would have suppressed the account, stated that Smith on the previous evening had arrived in London from his training quarters, and even went so far as to name the street he slept in. That Smith's visit to town was extremely injudicious there is not the slightest doubt, but this was not the worst. The time of departure for the scene of action was given, with a description of the place selected. This publicity had the effect of keeping away many who bad expressed a wish to be present, and on Slavin's side no less than eight gentlemen, each holding a high position in the sporting world, declined to have anything further to do with the affair. Who divulged the secret is at present a mystery, but there is not the slightest doubt that the information so freely put forward not only opened the eyes of the authorities, but gave the &quot;mob,&quot; to a certain extent, a key to the [Col. 5c–6a] situation. Slavin received the news with no small amount of astonishment, and a couple of telegrams informing him that a batch of the &quot;boys&quot; had gone across to make things warm for him did not give peace to his troubled mind. Smith made his way out of England on Friday by the short sea route, Dover to Calais, which he reached in safety. Slavin, not so fortunate, was arrested on Satuday, as already reported, but, after being bound over to keep the peace, he got away the same evening.&lt;/blockquote&gt;

==== Performance of the Men. ====
&lt;blockquote&gt;Jem Smith

was born in Whitecross-street, in the parish of Cripple-gate, on January 21, 1863. He stands 5ft 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in high, and weighs in condition 12st 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;?&lt;/sub&gt;in; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; hips, 41in; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; calf, 16&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing, after one or two rough-and-tumbles with his mates he was taken in hand by Jack Knifton (&quot;81-tonner&quot;), and later by Mr F. Grimm, of the Central Club. Smith has never been defeated, though in his first engagement he did not shape like a coming Champion. His first success was in a competition at the Griffin, Shoreditch. Later he beat Snowey, of Holloway, and was then matched to box Harry Arnold, with gloves, for a 10''l.'' purse. Arnold, though a stone and a half lighter than Smith, stripped in good condition, having for some time previously been engaged in training Jack Massey for an important engagement. The affair was looked upon as a good thing for Smith, but it took him nearly an hour to polish off the Seven Dials' representative, though he had slightly the best of the encounter from start to finish. Subsequently the veteran Goode deposited a [sov?] on behalf of Smith to match him to fight Massey, but this was forfeited. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for 25''l.'' a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end twelve rounds. Smith also won Tom Symonds's All-comers' Competition on February 26[?], 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. At the close Smith was presented with a large oil painting of himself, executed by a prominent patron of the sport. Smith's next battle was with Jack Davis for 100''l.'' a-side and the Championship, on the turf, and came off on the borders of Surrey and Sussex, on Wednesday, December 17. In all six rounds were fought, occupying 7min 56sec, Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for 300''l.'' at Maisonne-Lafitte, near Paris, on Tuesday, February 16, 1886 [?]. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Jack Knifton, the &quot;81-tonner,&quot; as he is called. This proved a ''fiasco'', a two days' dreary tramp on the Continent being followed by a return to the old country. Knifton's people claimed that Smith had more than the specified number of followers, but there is little doubt that only one side wanted a fight, and that was not the &quot;81-tonner's&quot; lot. Meeting after meeting was held, Smith offering to fight anywhere the other liked, and after much wrangling arrangements were made to bring the affair off near Oxted [?]. Somehow of other the police got wind of it, and overtaking the Knifton party near Croydon rendered another postponement necessary. this time the arrangements were left in the hands of Knifton's side, with the result that the principals, seconds, and a dozen others were captured early one morning in a furniture van by the police before they had gone fifty yards as their journey. All were brought up at Thames Police-court, and after a remand bound over to keep the peace for twelve months. The memorial struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19, 1887 [?]. Jem's recent performance was a Ten-round Glove Contest with Jack Wannop, the Champion laying 250''l.'' to 200''l.,'' and winning easily, following which he was beaten in two rounds at the Pelican Club by Peter Jackson, the Coloured Champion, in a glove contest, for 1000''l.''  [4, Col. 6b] 

Frank P. Slavin. 

the Champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft 1in in his stockings, and in condition scales 13st 5lb or 13st 6lb. Slavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. He always had a liking for the gloves, and sparred a great deal when young, but did not enter into any match until the latter part of 1885 — that is, for money, though he won several competitions previously as an amateur. Slavin's first fight was with knuckles against M. Power for 50''l.'' a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with 2oz gloves for 200''l.'' a-side and the Championship of Queensland, winning in three rounds. This was in December, 1885 [?], and in the following year the Australian formed a combination and went touring, offering purses ranging from 10''l.'' to 20''l.'' to any man who stood up for four rounds. At this game Slavin had an easy time of it; some stood one round, and some two, but none of them four. Towards the end of the year he went to Sydney, and entered into an agreement with Larry Foley, who has a big place there. Here Slavin met Taylor, a pupil of Jackson, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big fellow named Bligh for 10''l.'' a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for 100''l.'' a-side. Dooley had a big reputation, but Slavin won in nine rounds. The subject of these remarks then had a little difficulty in finding an opponent, but eventually Mike Costello, an American, a good man, made his appearance, and the pair were matched for 100''l.'' a-side. They had a big battle in Sydney, lasting two hours and a half, and a draw was the result. Another match was made for 400''l.'', and 50''l.'' a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for 24''l.'' a-side to stop Costello in six rounds, but the latter is a very clever and tricky man, and lasted the time. Subsequently Frank journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for 100''l.'' a=side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for 250''l'' a-side and the Championship. Jackson's party wanted a match with gloves for a purse. They could not come to terms, though Slavin's backers offered Jackson 50''l.'' to sign articles for 500''l.'' Australian was now a bit bare, so in January last year the Australian made a trip to New Zealand. The first man put up was Snath, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the Champion. Terms could not be agreed upon until they had &quot;fiddled&quot; Slavin for nearly six weeks. The pair were then matched for 200''l.'' a-side, and fought with 2oz gloves at [Wangazul?], Laing failing to come up at the end of the fifth round. After this Slavin went back to Melbourne, hoping to get on a match with Jackson, and was surprised to find he had gone to America. Following this he met Jack Burke, but after Slavin had had all the best of it the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for 200''l.'' with small gloves, which ended in Mike being knocked out in 2min 40sec. He was very bad, and frightened all by not coming to for nearly half an hour. Slavin returned to Melbourne, and Burke having put several challenges in the paper, our visitor offered to stop him in six rounds; 200''l.'' a-side was put up and articles signed for 4oz gloves, but on the night Burke's party threw down regular pillow-cases. Of course this did not suit the other side, and there was a fine old wrangle, lasting two hours and a half. Eventually Mr Joe Thompson, who was referee, declared all bets off, and the two fought 6oz gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for 200''l.'', with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. This is considered to be one of Slavin's best performances. About this time Woolf Bendoff, of Whitechapel, was in Melbourne, and finding plenty of backers, they put up a forfeit of 100''l.'' for the Londoner to box Slavin, for 500''l.'' a-side. After seeing the latter polish off Burke, however, Bendoff's friends would go no further with the atch, and Slavin accordingly pocketed the &quot;century.&quot; An easy victory over Bill Buck in two rounds followed, and then Slavin, on October 17 made his first appearance in England at Astley's Theatre, where he beat &quot;Chesterfield&quot; Goode in five rounds in a match of twelve rounds, for 400''l.''  [4, Col. 6c] &lt;/blockquote&gt;

==== THE PELICAN CLUB AND THE FIGHT. ====
&lt;blockquote&gt;To the Editor of the Sportsman. 

Sir, — As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it. — Yours truly,  

Cecil Raleigh, Hon. Sec. 

Pelican Club, 21, Denman-street, W., Dec. 23. 

__________ 

We hear on good authority that it is the intention of the members of the Pelican Club to present Slavin, the Australian Champion, with a belt, in recognition of the manly exhibition he gave yesterday at Bruges. &lt;/blockquote&gt;

==== THE CHAMPIONSHIP OF THE WORLD. ====
&lt;blockquote&gt;Frank Slavin to John L. Sullivan. 

Frank Slavin, the Champion of Australia, is prepared to meet John L. Sullivan, under London Prize Ring Rules, for 500''l.'' or 1000''l.'' a-side and the Championship of the World. Slavin has lodged with us a sum of 100''l.'', and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr R. K. Fox, of New York, and the money lodged with him or the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; will lead to a match.&lt;ref name=&quot;:1&quot; /&gt; &lt;/blockquote&gt;

=== Dundee ''Evening Telegraph'' ===
This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:&lt;blockquote&gt;There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.&lt;ref&gt;&quot;London Gives the Colonist a Great Reception.&quot; ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&amp;page=3&amp;article=052&amp;stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.&lt;blockquote&gt;The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; 

Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot;Police!&quot; Smith was hurried from the ring; but Slavin and his friends refused to budge, and, after &quot;Time&quot; had been called, they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to Margate. Smith and his friends were left behind at Bruges.

Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec.

Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other.

Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.)

Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him.

Round 5. — A light round, Smith going down before a light blow. Time, 11sec.

Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead.

Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners.

Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground.

Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min.

Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together.

Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander.

Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec.

Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin.

Round 14 and last. — Just a second before &quot;Time&quot; was called, Slavin, remarking on the treatment he had received, said, &quot;Are you Englishmen, boys?&quot; This was received with derisive shouts from Smith's party, and then, in an instant, a cry of &quot;Police&quot; was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, &quot;Why don't you give me fair play?&quot; Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, &quot;I shall refuse to act any longer. This man&quot; — pointing  to Slavin — &quot;can't have fair play. I declare it a draw.&quot; Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there.

Here the fight came to an end, the referee exclaiming, &quot;I shall refuse to act any longer. I shall declare it a draw.&quot; There was considerable surprise among Slavin and his supporters, and the Australian said, &quot;I came 16,000 miles to win, and why don't you give a man a chance?&quot; After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine.

Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— &quot;As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it.&quot; 

Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. 

That Slavin has now a right to the title (empty though it be) of &quot;champion&quot; is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— &quot;Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. 

Arrival of Slavin at Margate. 

Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, &quot;See the conquering hero comes.&quot; Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, &quot;Well, Slavin, how have you fared?&quot; &quot;Very badly,&quot; responded the Australian. &quot;As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders.&quot; — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. 

The Pelican Club and the Fight. 

A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— &quot;In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business.&quot; 

It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. 

The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— 

&quot;In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, &quot;F. P. SLAVIN. 

&quot;Nayland Rock Hotel, Margate.&quot; 

Mr. Fleming's Statement. 

Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. 

The Preliminary Arrangements. 

The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the &quot;lost art&quot; was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the &quot;old style.&quot; Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. 

There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. 

Performances of the Men. 

Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. 

Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; hips, 41in.; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; calf, 16&lt;sup&gt;3&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in.; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton (&quot;eighty-one tonner&quot;) and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the &quot;eighty-one-tonner&quot; as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.&lt;ref&gt;&quot;The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring.&quot; ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&amp;page=1&amp;article=006&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== The ''People'' ===

On Sunday, 29 December 1899, the ''People''&lt;blockquote&gt;Disgraceful Scenes. Expressions of indignation.

The prize fight between Jem Smith, who poses as the fistic &quot;champion of England,&quot; and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a &quot;strong corner.&quot; The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that &quot;Smith did not lose&quot; proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper.

The Fight.

The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot; Police!&quot; Smith was hurried from the ring, but Slavin and his friends refused to budge, and after &quot;Time&quot; had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of &quot;Foul,&quot; the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate.

Slavin's Opinion of his Treatment.

The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there.

Indignation in London.

Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee.

The Pelican Club Meeting.

At the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of &quot;the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.[&quot;] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed.

Slavin Challenges J. L. Sullivan.

Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&amp;page=16&amp;article=150&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Sportsman'' ===
On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the &quot;testimonial&quot; fund for Slavin as well as some letters to the editor about the Slavin-Smith match:&lt;blockquote&gt;
A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). 

The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris.

The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.'']

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Lord James Douglas  2 2 0
* Hon. Michael Sandys  2 2 0
* Fred Kotchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Proprietors of The Sportsman  2 2 0
* Mr. J. Purdey  2 2 0
* Mr. E. W. Bayly  2 2 0
* Messrs H. A. Domby and Son  2 2 0
* &quot;Caption Coe&quot;  2 2 0
* Mr C. W. Bulpett  2 2 0
* Mr P. W. Carr  2 2 0
* Mr W. Morley Pegge  2 2 0
* W. D. S.  2 2 0
* J. E. S.  2 2 0
* Mr W. H. Grenfel  2 2 0
* Mr Quartermaine East  2 2 0
* Mr H. H. Hungerford  2 2 0
* Mr Noel Fenwick  2 2 0
* Sir John Leslie, Bart.  2 2 0
* Mr T. Brinkman  2 0 0
* Mr Calder  1 1 0
* Mr H. W. Hill  1 1 0
* Mr John Young  1 1 0
* Mr D. A. Neilson  1 1 0
* Mr W. Neilson  1 1 0
* Mr Julius Simon  1 1 0
* Mr J. Edge  1 1 0
* Mr H. Marshall  1 1 0
* Mr F. Aynsley  1 1 0
* Col. Hon. Wellesley  1 1 0
* Mr Charles Hawtrey  1 1 0
* Mr W. F. Ashton  1 1 0
* Mr Barratt  1 1 0
* Old Durham Ox  1 0 0
* Sir John Willoughby  1 0 0
* A friend, per C. H. A.  0 10 6
* Mr Ben Giles  0 10 6
* Mr Tom Vale  0 10 6
* [E.?] W.  0 10 6
* Officer  0 10 0
* Per Mr Ike White, Exchange Club:
** S. W.  2 2 0
** Mr G. B. Knight  1 1 0
** Mr A. Powles  1 1 0
** Mr S. Blandford  1 1 0
** T. F. W. C.  1 1 0
** Mr Leo Harward  1 1 0
** Mr J. Harward  1 1 0
** Mr H. Harward  1 1 0
** Mr [Monteagle?] [Moutcagie??]  1 1 0
** Mr C. Orate  0 10 6
** W. G. P.  0 10 6
** H. R. T  0 10 6
** Mr Josiah Smith  0 10 6
** W. H. R.  0 10 6
** Mr R. Wilson  0 5 0
* Per Mr M. J. Harlow, Margate:
** Mr A. H. [Cosi? Coni?]  2 2 0
** Mr J. Thurston  1 1 0
** Mr J. A. Fagg  1 1 0
** Mr F. [Miles?]  1 1 0
** Mr Adolphe [Drineqbeir?]  1 1 0
** Mr Ernest [Drineqbier?]  1 1 0
** Mr Lacy Brown  1 1 0
** Mr Algy Brown  1 1 0
** Mr M. J. Harlow  1 1 0
** An Admirer  1 1 0
** Mr J. W. Robinson  1 1 0
** Mr W. Searles  1 1 0
** Mr J. Smith  1 1 0
** Mr J. Hume  1 1 0
** Mr J. Painter  1 1 0
** A Scotchman  1 1 0
** Mr J. Mundell  0 10 6
** Mr Bob Redman  0 10 6
** Mr G. A. Body  0 10 6
** Mr J. Phipps  0 10 6
** Mr J. Doughty  0 10 6
** Old Fisherman  0 10 6
** Mr Joe Knott  0 10 6
* Per Mr B. J. Angle
** Some mebers of the Stock Exchange  125 0 0
** Hr Henry J. Turner  2 2 0
* Per Mr W. J. Innes:
** Mr Sidney Fry  2 2 0
** Mr W. J. I.  2 2 0
** Mr W. Schlesinger  2 2 0
** Mr Joseph Thompson  2 2 0
** Mr Barney Thompson  2 2 0
** Mr L. Meyers  2 2 0
** Mr W. Shoe  2 2 0
** Mr George Cooper  2 2 0
** Mr J. T. Crossley  2 2 0
** Mr Andy Anderson  2 2 0
** Mr G. M. T.  2 2 0
** Mr F. R.  2 2 0
** Mr W. Bertram  1 1 0
** Mr J. W. Hammon  1 1 0
** Mr Alec Harris  1 1 0
** Mr John Isaacs  1 1 0
** Mr Edward [Dance? Dasce?]  1 1 0
** Mr George Blanchett  1 1 0
** Mr F. [Leleu? Lolou?]  1 1 0
** Mr W. J. R.  1 1 0
** Mr H. C.  1 1 0
** Mr A. A. M.  1 1 0
** Mr D. W. Fowler  0 10 6
* Per Mr F. H. Gale:
** Mr R. Moore  2 2 0
** Mr A. H. Goode  2 2 0
** Mr [B.? E.? R.?] J. G. Goode  2 2 0
** Mr F. H. Gale  2 2 0
** Mr H. Ulph  1 1 0
** Mr J. Hooper  1 1 0
** Mr C. Kittall  1 1 0
** Mr C. Moore  1 1 0
* Per Mr W. Riley:
** Mr Brand (Wiltshire)  2 2 0
** Mr Charles Godfrey  2 2 0
** Mrs Lane (Britannia)  2 2 0
** Mr Sam Adams  2 2 0
** B. and M.  0 7 6
* Per Mr A. Cockburn:
** Mr Richardson  2 2 0
** Mr A. Cox  1 1 0
** Mr Hy. Brackenbury  1 1 0

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and &quot;knuckle-dustered&quot; him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, &quot;Come and have a round.&quot; The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called &quot;seconds&quot; should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of &quot;Police&quot; was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska &quot;tough,&quot; as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how &quot;seconds&quot; — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a &quot;duster&quot; might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &amp;c.,

A Lover of Boxing.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &amp;c.,

Leicester, December 28. J. Barrett.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded.

I cannot, however, agree with those who hope &quot;that we have seen the last of prize-fighting,&quot; and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &amp;c.,

J. W. B. C., Junior Carlton Club, December 27.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself &quot;England's Champion.&quot; I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon &quot;British fair-play and courage.&quot; — Yours, a lover of it, Sydney Galvayne. 

York, December 26. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, 

Birmingham, December 27. Tom B. Sharp. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &amp;c., 

I. White.  

Exchange Club, Queen-street, Cheapside, Dec. 28.&lt;ref&gt;&quot;The Ring. The Slavin Testimonial.&quot; ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&amp;page=4&amp;article=020&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; &lt;/blockquote&gt;
On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:&lt;blockquote&gt;JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.
{{paragraph break}}

J&lt;small&gt;OHN&lt;/small&gt; L S&lt;small&gt;ULLIVAN&lt;/small&gt; has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me.&quot;&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:&lt;blockquote&gt;THE SLAVIN TESTIMONIAL
{{paragraph break}}
The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office.&lt;/blockquote&gt;
{| class=&quot;wikitable&quot;
!
!£
!s.
!d.
|-
|Amount already acknowledged
|407
|3
|6
|-
|Mr E.Tabernacle
|1
|1
|1
|-
|H. S.
|1
|1
|0
|}
&lt;blockquote&gt;The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Huddersfield Daily Chronicle'' ===
On 10 January 1890:&lt;blockquote&gt;
Mitchell and Lord Mandeville.

As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of &quot;sports&quot; at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by &quot;Pony&quot; Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &amp;c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. &quot; [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and &quot;went for&quot; Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and &quot;Pony&quot; Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was &quot;something up,&quot; and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&amp;page=4&amp;article=071&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;
After the Pelican Club meetings, Slavin comments:&lt;blockquote&gt;On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English &quot;champion,&quot; made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.&lt;ref&gt;&quot;Slavin at the Royal Acquarium.&quot; ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&amp;page=3&amp;article=042&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:&lt;blockquote&gt;Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. &quot;This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of &quot;heelers&quot; always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me.&quot;&lt;ref&gt;&quot;The Recent Prize Fight.&quot; ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&amp;page=6&amp;article=053&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Cork Constitution'' ===
About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:&lt;blockquote&gt;This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s &quot;lefts.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Colonial Standard and Jamaica Despatch'' ===
This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:&lt;blockquote&gt;The disgraceful affair, called by courtesy and [sic] &quot;International Prize Fight,&quot; which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The &quot;fight&quot; lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn [??, s/b Royston?], Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.&lt;ref&gt;&quot;London.&quot; ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&amp;page=2&amp;article=019&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;
== Notes and Questions ==
# 

==References==
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International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium.

==Overview==
On 23 December 1889 a championship boxing match in Bruges, Belgium, ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin or getting him near the ropes and beat him up, using fists, sticks, &quot;knuckle-dusters,&quot; &quot;life preservers&quot; and boots. The referee ruled the match a draw to prevent Slavin from being killed and then fled. The ''Cork Constitution'' called it &quot;the Smith-Slavin piece of brutality.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;

A knuckle-duster was &quot;A metal instrument made to cover the knuckles,&quot;&lt;ref&gt;“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.&lt;/ref&gt; like brass knuckles, and a life preserver was &quot;A stick or truncheon weighted at one end&quot; with lead.&lt;ref&gt;“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.&lt;/ref&gt;

One meeting of the boxing-oriented London Pelican Club was held almost immediately. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below).

Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England.

Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made here to offer complete coverage. Specialized sports coverage is privileged.

==Logistics==
* 23 December 1889, Monday, 8:00–9:00 a.m., actual fighting commencing at 8:35
* &quot;the private grounds of a gentleman's residence near Bruges, Belgium&quot;&lt;ref name=&quot;:1&quot;&gt;&quot;The Great Fight. Smith &amp; Slavin for £1000. Disorderly Proceedings. Brutal Attack on Slavin. Smith Knocked Down Repeatedly. He Leaves the Ring. A Draw Declared after 14 Rounds.&quot; ''The Sportsman'' 24 December 1889, Tuesday: 3 [of 4], Col. 1a – 4, Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&amp;page=3&amp;article=057&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; (3, Col. 1a)
* Stakes: £1,000
==Related Events==
* [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]]
* the &quot;[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]&quot; 
==Who Was Present==
# Jem Smith, British boxer
## Jack Harper, Smith's second
## Jack Baldock, Smith's second
## Mr J. Fleming, umpired for Smith
# Frank Slavin, Australian boxer
## Josh Cosnett, Slavin's second
## Bob Kirby, Slavin's second
## Mick Taylor, &quot;superintended the arrangements in his [Slavin's] corner&quot;
## Mr W. Riley umpired for Slavin
# &quot;the company round the ring
## Lord Mandeville — [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]]
## Lord Royston
## Hon. M. Sandys
## Mr Abington — Mr. G. Abington Baird, Smith's manager? and horseman (see description in Anthology)
## Mr F. Kotchie
## Mr J. Kilby — Jack Kilby, &quot;a real sportsman&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## Mr C. Parsons
## Mr Ike White
## Mr C. Morton
## Mr Harris
## Mr E. Bayley — Teddy Bayley
## Mr John Lewis (Australia) — Jack Lewis, Slavin's manager
## Mr J. Vesey — Joe Vesey, more often, or Vezey, referee
## Mr F. Hinde — Frank Hinde
## Mr M. Taylor
## Mr J. Calder — timekeeper
## &quot;Pop&quot; Joy
## R. Habbijam — Bob Habbijam
## G. (&quot;Barney&quot;) Sheppard — &quot;superintended the pitching of the ring&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3b)
## Jem Carney
## Tom Smith
## Mangie [?] Hyams
## Matthews
## Summers
## Boulton
## Mr G. T. Dunning — Tom Dunning, worked at ''The Sportsman'' and served as treasurer for the subscription to collect a cash prize for Slavin.
## &quot;several others, including two or three reporters&quot; — &quot;three scribes with pockets bulging from a surplusage of note-books dropped in&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## The writer of the big 24 December article, who reports for ''The Sportsman'' and who is not the &quot;well-known sporting journalist&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1b) who had heard that Slavin had been arrested in Margate
# &quot;an English resident in Bruges&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3a)
# &quot;Pop&quot; Hinde
# C. Horton
# &quot;a mob of Birmingham men&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
# Lord Rosslyn ?
# &quot;The Squire&quot;
#
==Entertainment==

==Anthology==

=== The Sportsman ===
The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 and most of page 4 (nearly 13 full columns) entirely to this story. This reporter's writing style is very ornate:

* &quot;Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping&quot; (3, Col. 1c)
* &quot;For a time it [the sea] was a bit rough, and one of the good sailors commenced quoting Byron. 'O'er the glad waters of the dark blue sea,' sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2a)
* &quot;Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2b)
* &quot;While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
* &quot;He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)

Besides the street-fighting terms (knuckle-dusters, life preservers) above, he also uses slang and local vocabulary like sabots (wooden clogs worn by working-class Belgians), Brums (people from Birmingham [?])

This is, then, a very long article, reproduced here in full except where the text is illegible, the very large block quote broken up by level-2 heads for navigation:&lt;blockquote&gt;The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below.&lt;/blockquote&gt;

==== THE JOURNEY TO THE CONTINENT. ====
&lt;blockquote&gt;(By our special reporter.)

[?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of &quot;globe-trotting,&quot; as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came]

A Rude Shock.

[Arriving??] at the office of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that &quot;something had happened.&quot; His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. &quot;What do you think of this?&quot; asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. &quot;Stoppage, but think can get bail. — Slavin.&quot; Here was a pretty &quot;go,&quot; the mystic words being translated meaning that the Australian had been &quot;pinched&quot; at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it

A May Morning.

Long before '''the earliest purveyor of the lacteal fluid''' arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of &quot;London peculiar&quot; had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking &quot;all broke up,&quot; put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a '''surplusage of note-books''' dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, &quot;Pop,&quot; and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey [sic], who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross &quot;The Squire,&quot; Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''.

Down to Dover.

It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly &quot;gave the game away,&quot; and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as &quot;dirty&quot; weather in store.

On the Briny.

The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and &quot;once on board the lugger&quot; my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the &quot;Princess&quot; swung off from the pier the two &quot;Bobs,&quot; Kirby and Habbijam, betook themselves to the &quot;rail&quot; in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. &quot;O'er the glad waters of the dark blue sea,&quot; sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having &quot;sought the seclusion which the cabin grants,&quot; and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in &quot;Pop,&quot; and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. [3, Col. 2a–2b] 

Ostend to Bruges.

While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded &quot;foreigner&quot; who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, &quot;Well, when is the 'mill' coming off?&quot; In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any &quot;mills.&quot; As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring '''that useful saponaceous compound''' with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity. [3, Col. 2b–2c]

Pugilism at the Piano.

While '''awaiting the refection''' Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The &quot;pelting of the pitiless storm&quot; continued the greater part of the night. 

Arrival of Jem Smith. 

The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arrangements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or &quot;the manger&quot; as &quot;Pop&quot; expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery &quot;Good morning, gentlemen,&quot; Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. 

The March to the Battlefield. 

Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out again. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work [3, Col. 3a–3b] 

Building the Ring. 

Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. 

The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, &quot;Pop&quot; Joy, R. Habbijam, G. (&quot;Barney&quot;) Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that 

The Dreaded Birmingham Division 

had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the &quot;boys&quot; worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, &quot;I hope all those present will keep quiet and help me all you can,&quot; at which brief speech there was a chorus of &quot;hear, hear,&quot; especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. 

The Call of &quot;Time&quot; 

At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, 

Smith was Knocked Down 

By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their &quot;Champion&quot; did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the &quot;Champion&quot; wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to &quot;be a man.&quot; Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of &quot;Here they are&quot; was raised, and 

A Stampede Ensued 

from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, &quot;There they are, there they are.&quot; By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, &quot;Are you Englishmen? I am an Englishman, and only want fair play.&quot; The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a &quot;duster,&quot; the referee saying, &quot;I saw that.&quot; As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, &quot;I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play.&quot; The referee concluded by giving the decision, &quot;It's a draw.&quot; The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting &quot;There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance.&quot; There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. 

There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended 

Making All Speed Homewards. 

The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. 

Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. &lt;/blockquote&gt;

==== THE FIGHT. ====
&lt;blockquote&gt;First Round. 

Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. 

Second Round. 

No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, &quot;First blood,&quot; a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On &quot;Time&quot; being called the pair were quick to scratch, that is to say, Slavin walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec.

Third Round.

Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking back to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec.

Fourth Round.

Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, 1min 15sec.

Fifth Round.

&quot;Don't go into his corner,&quot; urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite corner, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds.

Sixth Round.

Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec.

Seventh Round.

In vain did Slavin's friends implore him not to go into Smith's very &quot;warm corner.&quot; It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion was something terrible, and the ring looked very much like being invaded. The ropes in Smith's corner were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. 

Eighth Round.

Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth. Time of round, 1min 30sec.

Ninth Round.

This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a]

Tenth Round.

The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. 

Eleventh Round. 

Neither was quite so quick to time, and Harper, noticing that, said, &quot;This is not a two-round job, Patsey.&quot; Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to &quot;Come on and have a round.&quot; This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. 

Twelfth Round. 

A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. 

Thirteenth Round. 

Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, &quot;Oh!&quot; Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of &quot;Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a &quot;blue.&quot; The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words &quot;fight on&quot; came he and his friends were very delighted. 

Fourteenth and Last Round. 

When &quot;Time&quot; was called Slavin folded his arms and waited for Smith to &quot;come on,&quot; but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. 

No sooner was this round over than the referee called out. &quot;I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw.&quot; Then, going up to Slavin he said, &quot;I will not stay to see you killed.&quot; The Antipodean stood up and exclaimed, &quot;If you are Englishmen why don't you give me fair play? I have not had fair play.&quot; 

After this came a stampede. 

Remarks. 

Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as &quot;fights in the old style.&quot; That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the &quot;upper crust.&quot; Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being &quot;bashed&quot; on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with &quot;dusters&quot; and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a &quot;glover.&quot; With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a] that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or &quot;bottom&quot; of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, &quot;he was just getting warm.&quot; Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect.  

For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved.

From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play.

Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow. Bearing on this subject, Rule 26 says:— &quot;That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle.&quot; Rule 24 is also plain enough in all conscience, &quot;That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the &quot;boys&quot; and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but &quot;Kim&quot; soon showed he could hold his own with the &quot;boy,&quot; who beat a hasty retreat at the &quot;sight&quot; of six-inch &quot;Jack.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 1a)

We now take our leave of this unsavoury affair, which will probably be the last of its kind — &quot;a Championship knuckle fight.&quot; The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner &quot;merry mills in the old style&quot; die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office about three o'clock.

Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of &quot;dusters&quot; the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye.

It may be stated that not a single bet was made at the ring side. [4, Col. 1b]&lt;/blockquote&gt;

==== SLAVIN'S OPINION OF HIS TREATENT. ====
&lt;blockquote&gt;His Future Plans. 

On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. 

Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. 

With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. 

Much sympathy was expressed on all sides, and in the course of conversation with a S&lt;small&gt;PORTSMAN&lt;/small&gt; representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] 

Did you expect any trouble at the ring side? 

Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. 

Do you feel any effects of the fighting? 

Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. 

Why did you persist in going to his corner? 

Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. 

Of course you heard a mob was got together for Smith? 

Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. 

What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. [4, Col. 1c] 

I suppose you have a poor opinion of English fair play? 

Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. 

What are your plans for the future? 

Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. 

With the gloves? 

Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. 

You prefer the old style to the new? 

Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. 

Shall you give any exhibitions? 

Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. 

You put on weight the last week? 

Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. 

What do you think of our seconds? 

Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. 

When do you propose visiting London? 

Slavin: I'll come up to-morrow and call at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office about three. 

A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. &lt;/blockquote&gt;

==== A Testimonial for Slavin. ====
&lt;blockquote&gt;So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.'' The maximum donation was 2 guineas, which would be 2£ 2''s.''] 

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Hon. Michael Sandys  2 2 0
* Mr Fred Ketchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Mr J. Calder  2 2 0

Any contributions sent to the hon. treasurer will be duly acknowledged in T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;, which in a couple of days will contain the names of a committee now in course of formation. 

In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; 

Sir, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  &quot;Champion of England&quot; to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. 

If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C. 

Royal Thames Yacht Club, December 23. 

[We refer our correspondent to the paragraph above. — Ed. T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. ] [sic bracketed interpolation]  [4, Col. 2b] &lt;/blockquote&gt;

==== Eyewitnesses Interviewed. ====
&lt;blockquote&gt;A representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known &quot;sports&quot; (including more then one scion of a noble house) for a few minutes' chat. 

Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which did not number more than forty, being particularly noticeable. 

&quot;How did the combatants' seconds behave?&quot; — &quot;Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes.&quot; 

&quot;What form was the Englishman displaying?&quot; — &quot;From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds. Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy.&quot; 

&quot;How do you account for such a noisy corner at the back of Smith?&quot; — &quot;Although they did not number more than eight, they were some of the worst characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, inasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, &quot;framed&quot; [sic double quotes] up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his party.&quot; 

Are these men known at all?&quot; — &quot;Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period.&quot; 

&quot;Was not this sort of trouble expected?&quot; — &quot;It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated.&quot; 

In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. 

&quot;How did the crowd behave towards the finish?&quot; — &quot;There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious.&quot; 

&quot;Is it true that attempts were made to tamper with the water used by Slavin's seconds?&quot; — &quot;Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail.&quot; 

&quot;Did Smith deal out much punishment?&quot; — &quot;His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought for two hours more. He could have beaten Smith and both his seconds into the bargain, given fair play.&quot; 

&quot;Was Slavin injured during the fight?&quot; — &quot;No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed.&quot; 

&quot;As to the referee's decision?&quot; — &quot;That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened.&quot; 

&quot;What becamiof Smith after the fight.&quot; — &quot;He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does.&quot; 

&quot;How did the Cornstalk behave in the ring?&quot; — &quot;He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion.&quot; 

Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a &quot;mill,&quot; was very severe in his comments on the gang who were so lost to all sense of decency and fair play. 

Mr White said: — &quot;The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He had to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful.&quot;  [4, Col. 2c] 

&quot;Do you think it was for the best that the contest was stopped as it was?&quot; 

&quot;Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]&quot; 

&quot;What do you think of Slavin's abilities?&quot; 

&quot;He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt.&quot; 

On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. &quot;He is coming to town to-morrow,&quot; said the speaker, &quot;and shall have a great reception. Some good sportsmen have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it.&quot; 

&quot;What seemed the general opinion on the ground?&quot; 

&quot;Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took.&quot; &lt;/blockquote&gt;

==== Reception of the News in Birmingham. ====
&lt;blockquote&gt;Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a] and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. 

How the Match Was Made 

On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— &quot;He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, was unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers.&quot; [sic: quotation begins at the top of the paragraph] 

Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; and signed:   

Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', for the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally.   

In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names. [4, Col. 3c] &lt;br&gt;
Frank Hinde (for Frank Slavin). &lt;br&gt;
John Fleming (for James Smith). &lt;br&gt;
Witness for Smith, F. Montague. &lt;br&gt;
Witness for Stavin, John Lewes. &lt;br&gt;

Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The &quot;Cornstalk,&quot; though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not step in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds.   

Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell.   

Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[,] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice.
&lt;/blockquote&gt;

==== The Men in Training ====
&lt;blockquote&gt;Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp.   

Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound,  Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson.   &lt;/blockquote&gt;

==== Finding the Battle-Ground. ====
&lt;blockquote&gt;(By one of the Explorers.)   

At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class &quot;smoker&quot; attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed that a &quot;choppy sea&quot; was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. &quot;How confoundedly annoying,&quot;  he said. &quot;This is the sixth time I have been across during the year, and not once has it been calm.&quot; &quot;Better have a sleeping berth, sir,&quot; said a sprightly youth at the top of the stairs. &quot;Sleeping berth be hanged. I'll remain on deck,&quot; said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit &quot;peckish&quot; and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the &quot;magging&quot; [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, &quot;struck ile [?]&quot; with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The &quot;elderly party&quot; had a voice greatly resembling the dividing of tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. Just as I thought of giving up sleep for a bad job a gale sprang up. Though the Geselle is a record breaker she soon felt the effects of the breeze and began to roll heavily. The increase in the motion of the steamer was followed by a decrease in the conversation around me, and by degrees the punster went from had to worse. The &quot;[hussey??]&quot; [one?] also dropped into a walk, and after monosyllables had been exchanged the two came to a dead stop. Offering up a prayer of thanksgiving for this I closed my eyes again, and being a good sailor was quickly sleeping the peace of, I hope, the just. The concussion of the Gazelle against the Guernsey Pier, accompanied by a prod from Fleming's stick, at length aroused me.  [4, Col. 4b]   

Among the Channel Islands.   

The time by &quot;Joe Vickers&quot; was 7 a.m., and half-awake I took to my boots. &quot;Need not lace them; we shall be there in a minute,&quot; said Fleming, and in a drowsy, submissive, take-me-anywhere-if-only-to-bed sort of way I followed him on deck. A Babel of voices and a clutch at my bag and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a &quot;[sheenie?]&quot; (goodness knows what he meant, though), made us welcome, and agitating the communicator he handed us over to the care of a dark-eyed chambermaid. I forget her name, and as Fleming is not here I cannot give it. I will say no more than that after a cup of tea diluted with delicious cream, we took a well-earned and mucch needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the &quot;holiday maker&quot; and the &quot;honeymooner,&quot; is, as everybody knows, or should know, situate in the English Channel near the coasts of Brittany and Normandy. It is 68 [?] miles from Weymouth, 7 from Sark, 19 from Alderney [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in &quot;Mike.&quot; Said I, &quot;He's a Frenchman.&quot; &quot;Get out,&quot; replied Fleming, &quot;he's a 'bark.'&quot; &quot;Drinks he isn't,&quot; &quot;Done; drinks he is.&quot; &quot;Mike&quot; had not at that time spoken, but had noiselessly passed to and fro with a military step and a hand behind him. In reply to a call from Fleming he approached, and a &quot;Yes, yer honour,&quot; told me I should have to &quot;part.&quot; &quot;Mike&quot; was a trump, and I loved him from the start. For eleven years he had been in the service of a colonel at 1''l.'' a week, and three suit of clothes in the year. Hotel pay was not so good, and though he made &quot;a bit&quot; in the summer, the young ones were generally there when the bills were paid, if not when the work was on. &quot;Mike&quot; told us lots of things in what many consider the &quot;fascinating brogue.&quot; He informed us — and how our hearts beat at the news — that the island, which had a population of over 32,000, employed but six policemen. &quot;There's not a beggar in the land,&quot; said Mike, &quot;and but one Hansom cab, and,&quot; pointing to the window, &quot;be jabers, there it goes. &quot; So far so good, but after a walk round the son of Erin gave us a shock by saying, in reply to a query from Fleming, &quot;They've a way of their own on this island, and nearly ivery [sic] other man is a special constable.&quot; Fortunately we had just finished our repast, and, supporting each other, we tottered faintly to the door, partially regaining strength with a breath of the pure sea breeze. Beads of perspiration stood on the brow of the Pelican, whilst I was suffering from a &quot;trickling&quot; down the back of anything but a pleasing nature. Visions of a dungeon-cell under Castle Coronet flashed through my brain as a voice and a step were heard behind, and then I was somewhat reassured. &quot;When would you like a chair?&quot; fell softly on mine ear. Afraid that we had been thought of as a bit week on the leg, I turned to the speaker and asked what he meant. Explanations ensuing, we found the &quot;chair&quot; was a sort of Bath chair on wheels used throughout the island. Declining the offer, we took a stroll, and putting this and that together, came to the conclusion that our task was of a very delicate nature. The first thing was to interview the captain, who, in reply to a note, courteously waited upon us. He gave us the geography of the island and an introduction to Mr M., a friend, who we unfortunately found away from home. Taking a &quot;chair&quot; we then started on a voyage of exploration, making, as advised, our way to L'Ancresse [?] Common. Leaving St. Peter's Port and passing through St. Sampson's — the Vale — we came across acres of vineyards, in one instance extending nearly a mile along the road-side. Half an hour's drive brought us to the common mentioned, and dismounting we tried in vain to find a level piece of turf. Windmills abounded on all sides, and plodding on we reached L'Ancresse Bay, a lovely little spot, where the sea rippled softly over a bed of almost snow-white sand. Nowhere could be found a suitable spot whereon to pitch the &quot;magic square,&quot; and as the shades of night were falling fast we rattled back to the &quot;Royal.&quot; A wet night offered no inducement for further travel, and it was a case of early to bed and early to rise, the next day finding Fleming and myself at the door of Mr M. Acquainting this gentleman with the Nature of the business in hand he, like a good sportsman, fell into the scheme and at once volunteered to drive us to &quot;the best spot in the world.&quot; &quot;Bessie&quot; was harnessed to the village cart, and away we went at the rate of fully fourteen miles an hour in the direction of the Forest. I have sat behind many a good goer, but must for her size award the palm to Bessie a beauty barely twelve hands high. Throwing the reins in the bottom of the vehicle, Mr M. let the pony have her head, and the way she went uphill, down dale, and past other traps was truly astonishing. Great St. Peter's was passed, and lovely scenery encountered, the gardens being full of flowers, geraniums and roses blooming on all sides. An enjoyable drive of forty-five minutes found us at Boequaire [?] Bay, and directly in front stood the island of Lihou. As many of our readers may know, Lihou is a small hill of land that at low water can be reached on foot from Guernsey. As a fact, a paved roadway is laid in the bed of the ocean, across which both foot passengers and carts easily make their way. With the tide up communication is effected by boats, but this being low at the time we had the novel experience of   

Driving Through the Sea.   

Gaining the other side we &quot;explored&quot; and as the piece of land is only some three hundred yards by a hundred and fifty, we were not long about it. Two excellent patches of level turf were found, but just as we were making for the trap a gaunt, Robinson Crusoe-looking individual scaled one of the small stone walls and barred the way. Of course, he wanted to know our business, and, after a rather rough opening, quickly &quot;listened to reason.&quot; In the course of conversation we found that the population of the island was limited to three, the old &quot;josser&quot; and his two daughters, with a &quot;moke&quot; thrown in. Pointing a long skinny finger, which might have belonged to the Witch of Endor, towards the quadruped mentioned, the old man said, &quot;Us two keeps the family. He draws the seaweed and I burn it.&quot;  &quot;And why do you burn it?&quot; &quot;Well, to make [Col. 4c–5a] ashes, and from the ashes iodine is manufactured.&quot; &quot;And how much do yen earn a week by that?&quot; said  I. &quot;Oh, about 4''s.''; not 5''s.'' any week.&quot; Great Scott! and this was all the old man, his two daughters,  and the &quot;moke&quot; had to live on. Handing him a franc each, which nearly stupefied the old chappie, Fleming and myself returned to Mr M., who drove us rapidly back to St. Peter's Port. The next day we crossed to Sark and Jersey, but found nothing so suitable as the ground at Lihou, and fairly well satisfied with the result of our visit, a day later boarded the Lynx for the return journey.   

During our visit the weather had been of a fairly favourable description, but the voyage home was made under very depressing circumstances. Rain fell heavily and a thick mist hung over the water, which after going a few miles was alive with porpoise. Some of these animals came within a few yards of the Lynx, but the speed of the boat proved too good for the finney ones, who were eventually left behind. With a breeze springing up sail was hoisted, and dashing through a heavy sea in splendid form the Lynx made some sixteen knots in the hour. In this way half the journey was completed, when all of a sudden we were caught in a squall. In an instant the wind chopped round, and before you could say &quot;Jack Robinson&quot; the good vessel was turned broadside to. The order to take in sail was promptly obeyed, but the clatter on deck brought many an anxious passenger rapidly from below. A very heavy sea was now running, and though the Lynx kept her course bravely the decks were swept from stem to stern. A huge East India-man now hove in sight, and, as one of the men informed me, she had the audacity to cross our bows instead of passing under our stern. This breach of ocean etiquette ruffled the skipper of the Lynx, who chased the trader, and eventually made her alter her course. This little &quot;diversion&quot; caused much amusement amongst our passengers, and after a cold, rough passage we landed safely at Weymouth. Chilled to the bone, we made for the Victoria Hotel, where a chop and a glass of old October revived the almost stagnant circulation. London was reached about 9.30, and, tired and worn out, I was glad to lay my head on a pillow I had not seen for four days.   

From London to Ostend.

Unforseen [sic] difficulties having cropped up, the Channel Island scheme had to be abandoned; consequently we were once more called upon to leave our native land. This time Belgium was our destination, and having carefully studied the map of that country, we took our departure one fine starlight evening from Charing-cross by the 8.15 Continental express. In the previous week we had the luxury of a compartment to ourselves, but on the occasion under notice we were not so fortunate. Two individuals, one with a strong Midland accent, and the other a Frenchman speaking bad English, shared the carriage with us, and at such a rate did the former's tongue go directly he entered that I gave up reading as a bad job, with a sigh put away my unfinished book of the previous trip. Fleming fared better, and having a knack of ignoring all around him, once more tackled &quot;Blood Money,&quot; and almost read himself off to sleep.  In the meantime our friends in the opposite corner made &quot;hay,&quot; and I soon found out they were both in the lace trade. With a tongue that went a couple of hundred to the minute, the Midlander almost monopolised the conversation, from which I gleaned the two were bound for Calais, and thanked goodness we should part at Dover.

Reaching this port about ten we boarded the Ostend boat, which, as regards passengers, was very poorly patronised. Hardly a breath of wind was stirring, and with the sea calm as a mill-pond we had a pleasant passage over, and landed safely at Ostend soon after 3 a.m. A cup of coffee was hastily swallowed, and a half-hour's run in the Brussels express found us  [4, Col. 5b]

At Bruges.

Leaving the train, we had considerable difficulty in finding our way out of the station. For a moment we caught sight of a drowsy old porter, but the last carriage had not passed away from the platform ere he was lost to view. Seeing an opening on other side we made for it, only to have our passage barred by a half-dressed Belgian who, whilst displaying a total ignorance of English, Irish, and Scotch, directed us to return the way we came. For some time we tried in vain to find an outlet, and just as I had made up my mind to walk the platform until daylight Fleming found an iron gate ajar. Without loss of time we were in the street, and crossing the round, knocked loudly at the door of the nearest hotel. All was in darkness, and with no response to our fourth appeal for admittance, a return to the station or a &quot;tramp round&quot; appeared to be our only alternative. A slight noise inside followed by a shuffling of feet, again raised hope within us, and when, after some trouble, the door swung back and brough to view a shock-headed Belgian our relief was great. No time, I can assure you, was lost in getting between the sheets, and rising betimes we again went on a voyage of discovery. Fleming had a chum in the town, and to his house of call our steps were directed. Again disappointment was in store for our wanderers, and a feeling of disappointment came over me when we found Mr G. was away. There was, under the circumstances, nothing left but to wait, and Fleming, who puts on hair rapidly, decided to place himself under the care of the town barber, or, I should, properly speaking, my ''coiffeur''. For this piece of indiscretion he had to come away minus a slice of his lower lip, which, fortunately, was not large enough to seriously reduce his avordupois [sic?], and heaping blessings (?) [sic] on mankind generally, not in any way forgetting the &quot;Bruges Barber, we roamed over the old town, noted for its magnificent cathedral and many seats of learning. The hours sped on pleasantly enough, and in due time our friend was unearthed. Being pledged to secrecy I am not going from my word by giving full particulars of

The Battle Ground

placed at our disposal. This is in close proximity to the town — in fact, not half an hour's walk, and on reaching the spot we were shown a lovely piece of turn. A clump of trees on each side partly hid the house from view, and when I say that the ring was to be pitched on a lawn tennis ground, it will be seen that neither side could claim an advantage as regards standing ground, &amp;c., the lawn being as level as a billiard table. Though our efforts so far had been crowned with success, we did not get permission to use the &quot;pictch&quot; without much trouble. The matter had to be seriously considered, and Mr G. would not give a decided answer without waiting another day. Consequently our stay had to be extended and the following morning, after repeated promises that the affair should be conducted in an orderly manner with not a rough at the ring-side, we got the answer, &quot;Yes.&quot; &quot;But,&quot; said Mr G., &quot;if there is any row or crowd I can assure you you will have a journey for nothing; therefore if trouble is at all probable keep away and save your money and time.&quot; As long as faith was not broken by either side we saw no reason why the matter should not be settled in a proper manner, and making all the necessary arrangements and laying down plans of a most complete description, we returned to town well satisfied with the result of a visit to Bruges.  [4, Col. 5c]

Trouble about the Referee. 

On returning to town we received satisfactory reports of the work done by each of the contestants. Slavin, looked after by Josh Cosnett, the Birmingham heavy-weight, and his friend Mr Michael Taylor, of Australia, though rather light, was in good health. At one time he scaled only 12st 8lb, but a judicious spell of light work brought him back to 13st odd. Considering the importance of the business in hand, and its bearing on future events, Slavin had some  cause for anxiety. In addition to this he was fighting for 300''l.'' of his own money, and consequently wished to do himself justice in the international battle. 

The next question was the selection of someone capable and willing to fill the all-important post of referee. According to the articles it was left to the stakeholder to appoint the official named providing the representatives failed to agree, and in the end this course had to be adopted, solely owing to the objections lodged by Smith. The first gentleman suggested by Slavin's side was Mr B. J. Angle, whom Smith promptly objected to. Next a well-known owner of racehorses (a gentleman who adjudicated upon the knuckle fight between George Camp and &quot;Click&quot; Soles) was proposed with no better result. Mr Bat Murphy, strange to say, would not suit, and then in succession &quot;No&quot; was said to the names of  Messrs G. H. Vise, A. F. Bettinson, and [B.?] Bayley. All these gentlemen, it must be stated, were proposed by Slavin and objected to by Smith, who in the meantime had named a member of the Pelican Club, a gentleman of limited experience in these matters, and the landlord of the house from which he was training, Mr W. Riley, Newmarket. This brought about a deadlock, and the state of affairs getting wind, ugly rumours were soon afloat, and even money, and in many cases good odds, were betted that there would be &quot;no fight.&quot; All this while Slavin was chafing and fuming at the delay, whilst at the same time his  representative and Smith's were at work almost night and day endeavouring to bring about a settlement of the difficulty. For a long time it looked as though the matter would end in smoke, as after the services of the stakeholder had been called upon great difficulty was experienced in getting anyone to accept the office of referee. So late as Friday at noon nothing had been done, and it was not until the evening of that day that Mr Joe Vesey, consenting to stand, was nominated by the stakeholder. 

A Scare.

Though the battle-ground was only known to two persons, the country where the fight was to be decided being, in fact, or supposed to be, a perfect secret, on Friday morning a paragraph said to have been sent out by a news agency appeared in several papers, and created intense astonishment and consternation. This, set out in large type by those who we should have thought would have suppressed the account, stated that Smith on the previous evening had arrived in London from his training quarters, and even went so far as to name the street he slept in. That Smith's visit to town was extremely injudicious there is not the slightest doubt, but this was not the worst. The time of departure for the scene of action was given, with a description of the place selected. This publicity had the effect of keeping away many who bad expressed a wish to be present, and on Slavin's side no less than eight gentlemen, each holding a high position in the sporting world, declined to have anything further to do with the affair. Who divulged the secret is at present a mystery, but there is not the slightest doubt that the information so freely put forward not only opened the eyes of the authorities, but gave the &quot;mob,&quot; to a certain extent, a key to the [Col. 5c–6a] situation. Slavin received the news with no small amount of astonishment, and a couple of telegrams informing him that a batch of the &quot;boys&quot; had gone across to make things warm for him did not give peace to his troubled mind. Smith made his way out of England on Friday by the short sea route, Dover to Calais, which he reached in safety. Slavin, not so fortunate, was arrested on Satuday, as already reported, but, after being bound over to keep the peace, he got away the same evening.&lt;/blockquote&gt;

==== Performance of the Men. ====
&lt;blockquote&gt;Jem Smith

was born in Whitecross-street, in the parish of Cripple-gate, on January 21, 1863. He stands 5ft 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in high, and weighs in condition 12st 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;?&lt;/sub&gt;in; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; hips, 41in; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; calf, 16&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing, after one or two rough-and-tumbles with his mates he was taken in hand by Jack Knifton (&quot;81-tonner&quot;), and later by Mr F. Grimm, of the Central Club. Smith has never been defeated, though in his first engagement he did not shape like a coming Champion. His first success was in a competition at the Griffin, Shoreditch. Later he beat Snowey, of Holloway, and was then matched to box Harry Arnold, with gloves, for a 10''l.'' purse. Arnold, though a stone and a half lighter than Smith, stripped in good condition, having for some time previously been engaged in training Jack Massey for an important engagement. The affair was looked upon as a good thing for Smith, but it took him nearly an hour to polish off the Seven Dials' representative, though he had slightly the best of the encounter from start to finish. Subsequently the veteran Goode deposited a [sov?] on behalf of Smith to match him to fight Massey, but this was forfeited. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for 25''l.'' a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end twelve rounds. Smith also won Tom Symonds's All-comers' Competition on February 26[?], 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. At the close Smith was presented with a large oil painting of himself, executed by a prominent patron of the sport. Smith's next battle was with Jack Davis for 100''l.'' a-side and the Championship, on the turf, and came off on the borders of Surrey and Sussex, on Wednesday, December 17. In all six rounds were fought, occupying 7min 56sec, Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for 300''l.'' at Maisonne-Lafitte, near Paris, on Tuesday, February 16, 1886 [?]. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Jack Knifton, the &quot;81-tonner,&quot; as he is called. This proved a ''fiasco'', a two days' dreary tramp on the Continent being followed by a return to the old country. Knifton's people claimed that Smith had more than the specified number of followers, but there is little doubt that only one side wanted a fight, and that was not the &quot;81-tonner's&quot; lot. Meeting after meeting was held, Smith offering to fight anywhere the other liked, and after much wrangling arrangements were made to bring the affair off near Oxted [?]. Somehow of other the police got wind of it, and overtaking the Knifton party near Croydon rendered another postponement necessary. this time the arrangements were left in the hands of Knifton's side, with the result that the principals, seconds, and a dozen others were captured early one morning in a furniture van by the police before they had gone fifty yards as their journey. All were brought up at Thames Police-court, and after a remand bound over to keep the peace for twelve months. The memorial struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19, 1887 [?]. Jem's recent performance was a Ten-round Glove Contest with Jack Wannop, the Champion laying 250''l.'' to 200''l.,'' and winning easily, following which he was beaten in two rounds at the Pelican Club by Peter Jackson, the Coloured Champion, in a glove contest, for 1000''l.''  [4, Col. 6b] 

Frank P. Slavin. 

the Champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft 1in in his stockings, and in condition scales 13st 5lb or 13st 6lb. Slavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. He always had a liking for the gloves, and sparred a great deal when young, but did not enter into any match until the latter part of 1885 — that is, for money, though he won several competitions previously as an amateur. Slavin's first fight was with knuckles against M. Power for 50''l.'' a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with 2oz gloves for 200''l.'' a-side and the Championship of Queensland, winning in three rounds. This was in December, 1885 [?], and in the following year the Australian formed a combination and went touring, offering purses ranging from 10''l.'' to 20''l.'' to any man who stood up for four rounds. At this game Slavin had an easy time of it; some stood one round, and some two, but none of them four. Towards the end of the year he went to Sydney, and entered into an agreement with Larry Foley, who has a big place there. Here Slavin met Taylor, a pupil of Jackson, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big fellow named Bligh for 10''l.'' a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for 100''l.'' a-side. Dooley had a big reputation, but Slavin won in nine rounds. The subject of these remarks then had a little difficulty in finding an opponent, but eventually Mike Costello, an American, a good man, made his appearance, and the pair were matched for 100''l.'' a-side. They had a big battle in Sydney, lasting two hours and a half, and a draw was the result. Another match was made for 400''l.'', and 50''l.'' a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for 24''l.'' a-side to stop Costello in six rounds, but the latter is a very clever and tricky man, and lasted the time. Subsequently Frank journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for 100''l.'' a=side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for 250''l'' a-side and the Championship. Jackson's party wanted a match with gloves for a purse. They could not come to terms, though Slavin's backers offered Jackson 50''l.'' to sign articles for 500''l.'' Australian was now a bit bare, so in January last year the Australian made a trip to New Zealand. The first man put up was Snath, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the Champion. Terms could not be agreed upon until they had &quot;fiddled&quot; Slavin for nearly six weeks. The pair were then matched for 200''l.'' a-side, and fought with 2oz gloves at [Wangazul?], Laing failing to come up at the end of the fifth round. After this Slavin went back to Melbourne, hoping to get on a match with Jackson, and was surprised to find he had gone to America. Following this he met Jack Burke, but after Slavin had had all the best of it the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for 200''l.'' with small gloves, which ended in Mike being knocked out in 2min 40sec. He was very bad, and frightened all by not coming to for nearly half an hour. Slavin returned to Melbourne, and Burke having put several challenges in the paper, our visitor offered to stop him in six rounds; 200''l.'' a-side was put up and articles signed for 4oz gloves, but on the night Burke's party threw down regular pillow-cases. Of course this did not suit the other side, and there was a fine old wrangle, lasting two hours and a half. Eventually Mr Joe Thompson, who was referee, declared all bets off, and the two fought 6oz gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for 200''l.'', with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. This is considered to be one of Slavin's best performances. About this time Woolf Bendoff, of Whitechapel, was in Melbourne, and finding plenty of backers, they put up a forfeit of 100''l.'' for the Londoner to box Slavin, for 500''l.'' a-side. After seeing the latter polish off Burke, however, Bendoff's friends would go no further with the atch, and Slavin accordingly pocketed the &quot;century.&quot; An easy victory over Bill Buck in two rounds followed, and then Slavin, on October 17 made his first appearance in England at Astley's Theatre, where he beat &quot;Chesterfield&quot; Goode in five rounds in a match of twelve rounds, for 400''l.''  [4, Col. 6c] &lt;/blockquote&gt;

==== THE PELICAN CLUB AND THE FIGHT. ====
&lt;blockquote&gt;To the Editor of the Sportsman. 

Sir, — As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it. — Yours truly,  

Cecil Raleigh, Hon. Sec. 

Pelican Club, 21, Denman-street, W., Dec. 23. 

__________ 

We hear on good authority that it is the intention of the members of the Pelican Club to present Slavin, the Australian Champion, with a belt, in recognition of the manly exhibition he gave yesterday at Bruges. &lt;/blockquote&gt;

==== THE CHAMPIONSHIP OF THE WORLD. ====
&lt;blockquote&gt;Frank Slavin to John L. Sullivan. 

Frank Slavin, the Champion of Australia, is prepared to meet John L. Sullivan, under London Prize Ring Rules, for 500''l.'' or 1000''l.'' a-side and the Championship of the World. Slavin has lodged with us a sum of 100''l.'', and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr R. K. Fox, of New York, and the money lodged with him or the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; will lead to a match.&lt;ref name=&quot;:1&quot; /&gt; &lt;/blockquote&gt;

=== Dundee ''Evening Telegraph'' ===
This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:&lt;blockquote&gt;There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.&lt;ref&gt;&quot;London Gives the Colonist a Great Reception.&quot; ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&amp;page=3&amp;article=052&amp;stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.&lt;blockquote&gt;The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; 

Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot;Police!&quot; Smith was hurried from the ring; but Slavin and his friends refused to budge, and, after &quot;Time&quot; had been called, they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to Margate. Smith and his friends were left behind at Bruges.

Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec.

Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other.

Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.)

Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him.

Round 5. — A light round, Smith going down before a light blow. Time, 11sec.

Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead.

Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners.

Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground.

Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min.

Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together.

Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander.

Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec.

Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin.

Round 14 and last. — Just a second before &quot;Time&quot; was called, Slavin, remarking on the treatment he had received, said, &quot;Are you Englishmen, boys?&quot; This was received with derisive shouts from Smith's party, and then, in an instant, a cry of &quot;Police&quot; was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, &quot;Why don't you give me fair play?&quot; Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, &quot;I shall refuse to act any longer. This man&quot; — pointing  to Slavin — &quot;can't have fair play. I declare it a draw.&quot; Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there.

Here the fight came to an end, the referee exclaiming, &quot;I shall refuse to act any longer. I shall declare it a draw.&quot; There was considerable surprise among Slavin and his supporters, and the Australian said, &quot;I came 16,000 miles to win, and why don't you give a man a chance?&quot; After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine.

Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— &quot;As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it.&quot; 

Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. 

That Slavin has now a right to the title (empty though it be) of &quot;champion&quot; is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— &quot;Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. 

Arrival of Slavin at Margate. 

Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, &quot;See the conquering hero comes.&quot; Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, &quot;Well, Slavin, how have you fared?&quot; &quot;Very badly,&quot; responded the Australian. &quot;As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders.&quot; — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. 

The Pelican Club and the Fight. 

A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— &quot;In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business.&quot; 

It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. 

The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— 

&quot;In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, &quot;F. P. SLAVIN. 

&quot;Nayland Rock Hotel, Margate.&quot; 

Mr. Fleming's Statement. 

Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. 

The Preliminary Arrangements. 

The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the &quot;lost art&quot; was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the &quot;old style.&quot; Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. 

There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. 

Performances of the Men. 

Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. 

Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; hips, 41in.; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; calf, 16&lt;sup&gt;3&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in.; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton (&quot;eighty-one tonner&quot;) and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the &quot;eighty-one-tonner&quot; as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.&lt;ref&gt;&quot;The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring.&quot; ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&amp;page=1&amp;article=006&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== The ''People'' ===

On Sunday, 29 December 1899, the ''People''&lt;blockquote&gt;Disgraceful Scenes. Expressions of indignation.

The prize fight between Jem Smith, who poses as the fistic &quot;champion of England,&quot; and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a &quot;strong corner.&quot; The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that &quot;Smith did not lose&quot; proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper.

The Fight.

The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot; Police!&quot; Smith was hurried from the ring, but Slavin and his friends refused to budge, and after &quot;Time&quot; had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of &quot;Foul,&quot; the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate.

Slavin's Opinion of his Treatment.

The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there.

Indignation in London.

Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee.

The Pelican Club Meeting.

At the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of &quot;the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.[&quot;] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed.

Slavin Challenges J. L. Sullivan.

Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&amp;page=16&amp;article=150&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Sportsman'' ===
On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the &quot;testimonial&quot; fund for Slavin as well as some letters to the editor about the Slavin-Smith match:&lt;blockquote&gt;
A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). 

The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris.

The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.'']

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Lord James Douglas  2 2 0
* Hon. Michael Sandys  2 2 0
* Fred Kotchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Proprietors of The Sportsman  2 2 0
* Mr. J. Purdey  2 2 0
* Mr. E. W. Bayly  2 2 0
* Messrs H. A. Domby and Son  2 2 0
* &quot;Caption Coe&quot;  2 2 0
* Mr C. W. Bulpett  2 2 0
* Mr P. W. Carr  2 2 0
* Mr W. Morley Pegge  2 2 0
* W. D. S.  2 2 0
* J. E. S.  2 2 0
* Mr W. H. Grenfel  2 2 0
* Mr Quartermaine East  2 2 0
* Mr H. H. Hungerford  2 2 0
* Mr Noel Fenwick  2 2 0
* Sir John Leslie, Bart.  2 2 0
* Mr T. Brinkman  2 0 0
* Mr Calder  1 1 0
* Mr H. W. Hill  1 1 0
* Mr John Young  1 1 0
* Mr D. A. Neilson  1 1 0
* Mr W. Neilson  1 1 0
* Mr Julius Simon  1 1 0
* Mr J. Edge  1 1 0
* Mr H. Marshall  1 1 0
* Mr F. Aynsley  1 1 0
* Col. Hon. Wellesley  1 1 0
* Mr Charles Hawtrey  1 1 0
* Mr W. F. Ashton  1 1 0
* Mr Barratt  1 1 0
* Old Durham Ox  1 0 0
* Sir John Willoughby  1 0 0
* A friend, per C. H. A.  0 10 6
* Mr Ben Giles  0 10 6
* Mr Tom Vale  0 10 6
* [E.?] W.  0 10 6
* Officer  0 10 0
* Per Mr Ike White, Exchange Club:
** S. W.  2 2 0
** Mr G. B. Knight  1 1 0
** Mr A. Powles  1 1 0
** Mr S. Blandford  1 1 0
** T. F. W. C.  1 1 0
** Mr Leo Harward  1 1 0
** Mr J. Harward  1 1 0
** Mr H. Harward  1 1 0
** Mr [Monteagle?] [Moutcagie??]  1 1 0
** Mr C. Orate  0 10 6
** W. G. P.  0 10 6
** H. R. T  0 10 6
** Mr Josiah Smith  0 10 6
** W. H. R.  0 10 6
** Mr R. Wilson  0 5 0
* Per Mr M. J. Harlow, Margate:
** Mr A. H. [Cosi? Coni?]  2 2 0
** Mr J. Thurston  1 1 0
** Mr J. A. Fagg  1 1 0
** Mr F. [Miles?]  1 1 0
** Mr Adolphe [Drineqbeir?]  1 1 0
** Mr Ernest [Drineqbier?]  1 1 0
** Mr Lacy Brown  1 1 0
** Mr Algy Brown  1 1 0
** Mr M. J. Harlow  1 1 0
** An Admirer  1 1 0
** Mr J. W. Robinson  1 1 0
** Mr W. Searles  1 1 0
** Mr J. Smith  1 1 0
** Mr J. Hume  1 1 0
** Mr J. Painter  1 1 0
** A Scotchman  1 1 0
** Mr J. Mundell  0 10 6
** Mr Bob Redman  0 10 6
** Mr G. A. Body  0 10 6
** Mr J. Phipps  0 10 6
** Mr J. Doughty  0 10 6
** Old Fisherman  0 10 6
** Mr Joe Knott  0 10 6
* Per Mr B. J. Angle
** Some mebers of the Stock Exchange  125 0 0
** Hr Henry J. Turner  2 2 0
* Per Mr W. J. Innes:
** Mr Sidney Fry  2 2 0
** Mr W. J. I.  2 2 0
** Mr W. Schlesinger  2 2 0
** Mr Joseph Thompson  2 2 0
** Mr Barney Thompson  2 2 0
** Mr L. Meyers  2 2 0
** Mr W. Shoe  2 2 0
** Mr George Cooper  2 2 0
** Mr J. T. Crossley  2 2 0
** Mr Andy Anderson  2 2 0
** Mr G. M. T.  2 2 0
** Mr F. R.  2 2 0
** Mr W. Bertram  1 1 0
** Mr J. W. Hammon  1 1 0
** Mr Alec Harris  1 1 0
** Mr John Isaacs  1 1 0
** Mr Edward [Dance? Dasce?]  1 1 0
** Mr George Blanchett  1 1 0
** Mr F. [Leleu? Lolou?]  1 1 0
** Mr W. J. R.  1 1 0
** Mr H. C.  1 1 0
** Mr A. A. M.  1 1 0
** Mr D. W. Fowler  0 10 6
* Per Mr F. H. Gale:
** Mr R. Moore  2 2 0
** Mr A. H. Goode  2 2 0
** Mr [B.? E.? R.?] J. G. Goode  2 2 0
** Mr F. H. Gale  2 2 0
** Mr H. Ulph  1 1 0
** Mr J. Hooper  1 1 0
** Mr C. Kittall  1 1 0
** Mr C. Moore  1 1 0
* Per Mr W. Riley:
** Mr Brand (Wiltshire)  2 2 0
** Mr Charles Godfrey  2 2 0
** Mrs Lane (Britannia)  2 2 0
** Mr Sam Adams  2 2 0
** B. and M.  0 7 6
* Per Mr A. Cockburn:
** Mr Richardson  2 2 0
** Mr A. Cox  1 1 0
** Mr Hy. Brackenbury  1 1 0

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and &quot;knuckle-dustered&quot; him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, &quot;Come and have a round.&quot; The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called &quot;seconds&quot; should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of &quot;Police&quot; was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska &quot;tough,&quot; as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how &quot;seconds&quot; — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a &quot;duster&quot; might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &amp;c.,

A Lover of Boxing.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &amp;c.,

Leicester, December 28. J. Barrett.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded.

I cannot, however, agree with those who hope &quot;that we have seen the last of prize-fighting,&quot; and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &amp;c.,

J. W. B. C., Junior Carlton Club, December 27.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself &quot;England's Champion.&quot; I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon &quot;British fair-play and courage.&quot; — Yours, a lover of it, Sydney Galvayne. 

York, December 26. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, 

Birmingham, December 27. Tom B. Sharp. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &amp;c., 

I. White.  

Exchange Club, Queen-street, Cheapside, Dec. 28.&lt;ref&gt;&quot;The Ring. The Slavin Testimonial.&quot; ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&amp;page=4&amp;article=020&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; &lt;/blockquote&gt;
On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:&lt;blockquote&gt;JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.
{{paragraph break}}

J&lt;small&gt;OHN&lt;/small&gt; L S&lt;small&gt;ULLIVAN&lt;/small&gt; has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me.&quot;&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:&lt;blockquote&gt;THE SLAVIN TESTIMONIAL
{{paragraph break}}
The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office.&lt;/blockquote&gt;
{| class=&quot;wikitable&quot;
!
!£
!s.
!d.
|-
|Amount already acknowledged
|407
|3
|6
|-
|Mr E.Tabernacle
|1
|1
|1
|-
|H. S.
|1
|1
|0
|}
&lt;blockquote&gt;The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Huddersfield Daily Chronicle'' ===
On 10 January 1890:&lt;blockquote&gt;
Mitchell and Lord Mandeville.

As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of &quot;sports&quot; at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by &quot;Pony&quot; Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &amp;c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. &quot; [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and &quot;went for&quot; Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and &quot;Pony&quot; Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was &quot;something up,&quot; and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&amp;page=4&amp;article=071&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;
After the Pelican Club meetings, Slavin comments:&lt;blockquote&gt;On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English &quot;champion,&quot; made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.&lt;ref&gt;&quot;Slavin at the Royal Acquarium.&quot; ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&amp;page=3&amp;article=042&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:&lt;blockquote&gt;Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. &quot;This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of &quot;heelers&quot; always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me.&quot;&lt;ref&gt;&quot;The Recent Prize Fight.&quot; ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&amp;page=6&amp;article=053&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Cork Constitution'' ===
About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:&lt;blockquote&gt;This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s &quot;lefts.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Colonial Standard and Jamaica Despatch'' ===
This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:&lt;blockquote&gt;The disgraceful affair, called by courtesy and [sic] &quot;International Prize Fight,&quot; which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The &quot;fight&quot; lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn [??, s/b Royston?], Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.&lt;ref&gt;&quot;London.&quot; ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&amp;page=2&amp;article=019&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;
== Notes and Questions ==
# 

==References==
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        <id>1331941</id>
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      <text bytes="188857" sha1="jxehmxqe9iyccxcavc2gvbdpra4pr4b" xml:space="preserve">=Event=
International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium.

==Overview==
On 23 December 1889 a championship boxing match in Bruges, Belgium, ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin or getting him near the ropes and beat him up, using fists, sticks, &quot;knuckle-dusters,&quot; &quot;life preservers&quot; and boots. The referee ruled the match a draw to prevent Slavin from being killed and then fled. The ''Cork Constitution'' called it &quot;the Smith-Slavin piece of brutality.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;

A knuckle-duster was &quot;A metal instrument made to cover the knuckles,&quot;&lt;ref&gt;“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.&lt;/ref&gt; like brass knuckles, and a life preserver was &quot;A stick or truncheon weighted at one end&quot; with lead.&lt;ref&gt;“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.&lt;/ref&gt;

One meeting of the boxing-oriented London Pelican Club was held almost immediately. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below).

Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England.

Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made here to offer complete coverage. Specialized sports coverage is privileged.

==Logistics==
* 23 December 1889, Monday, 8:00–9:00 a.m., actual fighting commencing at 8:35
* &quot;the private grounds of a gentleman's residence near Bruges, Belgium&quot;&lt;ref name=&quot;:1&quot;&gt;&quot;The Great Fight. Smith &amp; Slavin for £1000. Disorderly Proceedings. Brutal Attack on Slavin. Smith Knocked Down Repeatedly. He Leaves the Ring. A Draw Declared after 14 Rounds.&quot; ''The Sportsman'' 24 December 1889, Tuesday: 3 [of 4], Col. 1a – 4, Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&amp;page=3&amp;article=057&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; (3, Col. 1a)
* Stakes: £1,000
==Related Events==
* [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]]
* the &quot;[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]&quot; 
==Who Was Present==
# Jem Smith, British boxer
## Jack Harper, Smith's second
## Jack Baldock, Smith's second
## Mr J. Fleming, umpired for Smith
# Frank Slavin, Australian boxer
## Josh Cosnett, Slavin's second
## Bob Kirby, Slavin's second
## Mick Taylor, &quot;superintended the arrangements in his [Slavin's] corner&quot;
## Mr W. Riley umpired for Slavin
# &quot;the company round the ring
## Lord Mandeville — [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]]
## Lord Royston
## Hon. M. Sandys
## Mr Abington — Mr. G. Abington Baird, Smith's manager? and horseman (see description in Anthology)
## Mr F. Kotchie
## Mr J. Kilby — Jack Kilby, &quot;a real sportsman&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## Mr C. Parsons
## Mr Ike White
## Mr C. Morton
## Mr Harris
## Mr E. Bayley — Teddy Bayley
## Mr John Lewis (Australia) — Jack Lewis, Slavin's manager
## Mr J. Vesey — Joe Vesey, more often, or Vezey, referee
## Mr F. Hinde — Frank Hinde
## Mr M. Taylor
## Mr J. Calder — timekeeper
## &quot;Pop&quot; Joy
## R. Habbijam — Bob Habbijam
## G. (&quot;Barney&quot;) Sheppard — &quot;superintended the pitching of the ring&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3b)
## Jem Carney
## Tom Smith
## Mangie [?] Hyams
## Matthews
## Summers
## Boulton
## Mr G. T. Dunning — Tom Dunning, worked at ''The Sportsman'' and served as treasurer for the subscription to collect a cash prize for Slavin.
## &quot;several others, including two or three reporters&quot; — &quot;three scribes with pockets bulging from a surplusage of note-books dropped in&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## The writer of the big 24 December article, who reports for ''The Sportsman'' and who is not the &quot;well-known sporting journalist&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1b) who had heard that Slavin had been arrested in Margate
# &quot;an English resident in Bruges&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3a)
# &quot;Pop&quot; Hinde
# C. Horton
# &quot;a mob of Birmingham men&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
# Lord Rosslyn ?
# &quot;The Squire&quot;
#
==Entertainment==

==Anthology==

=== The Sportsman ===
The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 and most of page 4 (nearly 13 full columns) entirely to this story. This reporter's writing style is very ornate:

* &quot;Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping&quot; (3, Col. 1c)
* &quot;For a time it [the sea] was a bit rough, and one of the good sailors commenced quoting Byron. 'O'er the glad waters of the dark blue sea,' sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2a)
* &quot;Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2b)
* &quot;While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
* &quot;He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)

Besides the street-fighting terms (knuckle-dusters, life preservers) above, he also uses slang and local vocabulary like sabots (wooden clogs worn by working-class Belgians), Brums (people from Birmingham [?])

This is, then, a very long article, reproduced here in full except where the text is illegible, the very large block quote broken up by level-2 heads for navigation:&lt;blockquote&gt;The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below.&lt;/blockquote&gt;

==== THE JOURNEY TO THE CONTINENT. ====
&lt;blockquote&gt;(By our special reporter.)

[?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of &quot;globe-trotting,&quot; as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came]

A Rude Shock.

[Arriving??] at the office of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that &quot;something had happened.&quot; His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. &quot;What do you think of this?&quot; asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. &quot;Stoppage, but think can get bail. — Slavin.&quot; Here was a pretty &quot;go,&quot; the mystic words being translated meaning that the Australian had been &quot;pinched&quot; at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it

A May Morning.

Long before '''the earliest purveyor of the lacteal fluid''' arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of &quot;London peculiar&quot; had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking &quot;all broke up,&quot; put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a '''surplusage of note-books''' dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, &quot;Pop,&quot; and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey [sic], who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross &quot;The Squire,&quot; Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''.

Down to Dover.

It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly &quot;gave the game away,&quot; and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as &quot;dirty&quot; weather in store.

On the Briny.

The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and &quot;once on board the lugger&quot; my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the &quot;Princess&quot; swung off from the pier the two &quot;Bobs,&quot; Kirby and Habbijam, betook themselves to the &quot;rail&quot; in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. &quot;O'er the glad waters of the dark blue sea,&quot; sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having &quot;sought the seclusion which the cabin grants,&quot; and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in &quot;Pop,&quot; and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. [3, Col. 2a–2b] 

Ostend to Bruges.

While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded &quot;foreigner&quot; who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, &quot;Well, when is the 'mill' coming off?&quot; In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any &quot;mills.&quot; As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring '''that useful saponaceous compound''' with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity. [3, Col. 2b–2c]

Pugilism at the Piano.

While '''awaiting the refection''' Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The &quot;pelting of the pitiless storm&quot; continued the greater part of the night. 

Arrival of Jem Smith. 

The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arrangements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or &quot;the manger&quot; as &quot;Pop&quot; expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery &quot;Good morning, gentlemen,&quot; Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. 

The March to the Battlefield. 

Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out again. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work [3, Col. 3a–3b] 

Building the Ring. 

Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. 

The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, &quot;Pop&quot; Joy, R. Habbijam, G. (&quot;Barney&quot;) Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that 

The Dreaded Birmingham Division 

had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the &quot;boys&quot; worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, &quot;I hope all those present will keep quiet and help me all you can,&quot; at which brief speech there was a chorus of &quot;hear, hear,&quot; especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. 

The Call of &quot;Time&quot; 

At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, 

Smith was Knocked Down 

By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their &quot;Champion&quot; did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the &quot;Champion&quot; wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to &quot;be a man.&quot; Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of &quot;Here they are&quot; was raised, and 

A Stampede Ensued 

from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, &quot;There they are, there they are.&quot; By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, &quot;Are you Englishmen? I am an Englishman, and only want fair play.&quot; The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a &quot;duster,&quot; the referee saying, &quot;I saw that.&quot; As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, &quot;I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play.&quot; The referee concluded by giving the decision, &quot;It's a draw.&quot; The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting &quot;There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance.&quot; There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. 

There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended 

Making All Speed Homewards. 

The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. 

Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. &lt;/blockquote&gt;

==== THE FIGHT. ====
&lt;blockquote&gt;First Round. 

Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. 

Second Round. 

No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, &quot;First blood,&quot; a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On &quot;Time&quot; being called the pair were quick to scratch, that is to say, Slavin walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec.

Third Round.

Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking back to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec.

Fourth Round.

Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, 1min 15sec.

Fifth Round.

&quot;Don't go into his corner,&quot; urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite corner, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds.

Sixth Round.

Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec.

Seventh Round.

In vain did Slavin's friends implore him not to go into Smith's very &quot;warm corner.&quot; It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion was something terrible, and the ring looked very much like being invaded. The ropes in Smith's corner were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. 

Eighth Round.

Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth. Time of round, 1min 30sec.

Ninth Round.

This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a]

Tenth Round.

The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. 

Eleventh Round. 

Neither was quite so quick to time, and Harper, noticing that, said, &quot;This is not a two-round job, Patsey.&quot; Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to &quot;Come on and have a round.&quot; This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. 

Twelfth Round. 

A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. 

Thirteenth Round. 

Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, &quot;Oh!&quot; Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of &quot;Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a &quot;blue.&quot; The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words &quot;fight on&quot; came he and his friends were very delighted. 

Fourteenth and Last Round. 

When &quot;Time&quot; was called Slavin folded his arms and waited for Smith to &quot;come on,&quot; but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. 

No sooner was this round over than the referee called out. &quot;I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw.&quot; Then, going up to Slavin he said, &quot;I will not stay to see you killed.&quot; The Antipodean stood up and exclaimed, &quot;If you are Englishmen why don't you give me fair play? I have not had fair play.&quot; 

After this came a stampede. 

Remarks. 

Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as &quot;fights in the old style.&quot; That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the &quot;upper crust.&quot; Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being &quot;bashed&quot; on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with &quot;dusters&quot; and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a &quot;glover.&quot; With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a] that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or &quot;bottom&quot; of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, &quot;he was just getting warm.&quot; Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect.  

For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved.

From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play.

Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow. Bearing on this subject, Rule 26 says:— &quot;That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle.&quot; Rule 24 is also plain enough in all conscience, &quot;That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the &quot;boys&quot; and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but &quot;Kim&quot; soon showed he could hold his own with the &quot;boy,&quot; who beat a hasty retreat at the &quot;sight&quot; of six-inch &quot;Jack.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 1a)

We now take our leave of this unsavoury affair, which will probably be the last of its kind — &quot;a Championship knuckle fight.&quot; The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner &quot;merry mills in the old style&quot; die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office about three o'clock.

Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of &quot;dusters&quot; the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye.

It may be stated that not a single bet was made at the ring side. [4, Col. 1b]&lt;/blockquote&gt;

==== SLAVIN'S OPINION OF HIS TREATENT. ====
&lt;blockquote&gt;His Future Plans. 

On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. 

Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. 

With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. 

Much sympathy was expressed on all sides, and in the course of conversation with a S&lt;small&gt;PORTSMAN&lt;/small&gt; representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] 

Did you expect any trouble at the ring side? 

Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. 

Do you feel any effects of the fighting? 

Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. 

Why did you persist in going to his corner? 

Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. 

Of course you heard a mob was got together for Smith? 

Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. 

What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. [4, Col. 1c] 

I suppose you have a poor opinion of English fair play? 

Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. 

What are your plans for the future? 

Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. 

With the gloves? 

Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. 

You prefer the old style to the new? 

Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. 

Shall you give any exhibitions? 

Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. 

You put on weight the last week? 

Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. 

What do you think of our seconds? 

Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. 

When do you propose visiting London? 

Slavin: I'll come up to-morrow and call at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office about three. 

A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. &lt;/blockquote&gt;

==== A Testimonial for Slavin. ====
&lt;blockquote&gt;So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.'' The maximum donation was 2 guineas, which would be 2£ 2''s.''] 

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Hon. Michael Sandys  2 2 0
* Mr Fred Ketchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Mr J. Calder  2 2 0

Any contributions sent to the hon. treasurer will be duly acknowledged in T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;, which in a couple of days will contain the names of a committee now in course of formation. 

In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; 

Sir, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  &quot;Champion of England&quot; to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. 

If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C.&lt;br&gt;
Royal Thames Yacht Club, December 23. 

[We refer our correspondent to the paragraph above. — Ed. T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. ] [sic bracketed interpolation]  [4, Col. 2b] &lt;/blockquote&gt;

==== Eyewitnesses Interviewed. ====
&lt;blockquote&gt;A representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known &quot;sports&quot; (including more then one scion of a noble house) for a few minutes' chat. 

Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which did not number more than forty, being particularly noticeable. 

&quot;How did the combatants' seconds behave?&quot; — &quot;Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes.&quot; 

&quot;What form was the Englishman displaying?&quot; — &quot;From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds. Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy.&quot; 

&quot;How do you account for such a noisy corner at the back of Smith?&quot; — &quot;Although they did not number more than eight, they were some of the worst characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, inasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, &quot;framed&quot; [sic double quotes] up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his party.&quot; 

Are these men known at all?&quot; — &quot;Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period.&quot; 

&quot;Was not this sort of trouble expected?&quot; — &quot;It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated.&quot; 

In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. 

&quot;How did the crowd behave towards the finish?&quot; — &quot;There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious.&quot; 

&quot;Is it true that attempts were made to tamper with the water used by Slavin's seconds?&quot; — &quot;Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail.&quot; 

&quot;Did Smith deal out much punishment?&quot; — &quot;His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought for two hours more. He could have beaten Smith and both his seconds into the bargain, given fair play.&quot; 

&quot;Was Slavin injured during the fight?&quot; — &quot;No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed.&quot; 

&quot;As to the referee's decision?&quot; — &quot;That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened.&quot; 

&quot;What becamiof Smith after the fight.&quot; — &quot;He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does.&quot; 

&quot;How did the Cornstalk behave in the ring?&quot; — &quot;He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion.&quot; 

Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a &quot;mill,&quot; was very severe in his comments on the gang who were so lost to all sense of decency and fair play. 

Mr White said: — &quot;The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He had to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful.&quot;  [4, Col. 2c] 

&quot;Do you think it was for the best that the contest was stopped as it was?&quot; 

&quot;Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]&quot; 

&quot;What do you think of Slavin's abilities?&quot; 

&quot;He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt.&quot; 

On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. &quot;He is coming to town to-morrow,&quot; said the speaker, &quot;and shall have a great reception. Some good sportsmen have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it.&quot; 

&quot;What seemed the general opinion on the ground?&quot; 

&quot;Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took.&quot; &lt;/blockquote&gt;

==== Reception of the News in Birmingham. ====
&lt;blockquote&gt;Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a] and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. 

How the Match Was Made 

On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— &quot;He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, was unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers.&quot; [sic: quotation begins at the top of the paragraph] 

Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; and signed:   

Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', for the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally.   

In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names. [4, Col. 3c] &lt;br&gt;
Frank Hinde (for Frank Slavin). &lt;br&gt;
John Fleming (for James Smith). &lt;br&gt;
Witness for Smith, F. Montague. &lt;br&gt;
Witness for Stavin, John Lewes. &lt;br&gt;

Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The &quot;Cornstalk,&quot; though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not step in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds.   

Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell.   

Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[,] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice.
&lt;/blockquote&gt;

==== The Men in Training ====
&lt;blockquote&gt;Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp.   

Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound,  Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson.   &lt;/blockquote&gt;

==== Finding the Battle-Ground. ====
&lt;blockquote&gt;(By one of the Explorers.)   

At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class &quot;smoker&quot; attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed that a &quot;choppy sea&quot; was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. &quot;How confoundedly annoying,&quot;  he said. &quot;This is the sixth time I have been across during the year, and not once has it been calm.&quot; &quot;Better have a sleeping berth, sir,&quot; said a sprightly youth at the top of the stairs. &quot;Sleeping berth be hanged. I'll remain on deck,&quot; said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit &quot;peckish&quot; and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the &quot;magging&quot; [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, &quot;struck ile [?]&quot; with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The &quot;elderly party&quot; had a voice greatly resembling the dividing of tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. Just as I thought of giving up sleep for a bad job a gale sprang up. Though the Geselle is a record breaker she soon felt the effects of the breeze and began to roll heavily. The increase in the motion of the steamer was followed by a decrease in the conversation around me, and by degrees the punster went from had to worse. The &quot;[hussey??]&quot; [one?] also dropped into a walk, and after monosyllables had been exchanged the two came to a dead stop. Offering up a prayer of thanksgiving for this I closed my eyes again, and being a good sailor was quickly sleeping the peace of, I hope, the just. The concussion of the Gazelle against the Guernsey Pier, accompanied by a prod from Fleming's stick, at length aroused me.  [4, Col. 4b]   

Among the Channel Islands.   

The time by &quot;Joe Vickers&quot; was 7 a.m., and half-awake I took to my boots. &quot;Need not lace them; we shall be there in a minute,&quot; said Fleming, and in a drowsy, submissive, take-me-anywhere-if-only-to-bed sort of way I followed him on deck. A Babel of voices and a clutch at my bag and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a &quot;[sheenie?]&quot; (goodness knows what he meant, though), made us welcome, and agitating the communicator he handed us over to the care of a dark-eyed chambermaid. I forget her name, and as Fleming is not here I cannot give it. I will say no more than that after a cup of tea diluted with delicious cream, we took a well-earned and mucch needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the &quot;holiday maker&quot; and the &quot;honeymooner,&quot; is, as everybody knows, or should know, situate in the English Channel near the coasts of Brittany and Normandy. It is 68 [?] miles from Weymouth, 7 from Sark, 19 from Alderney [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in &quot;Mike.&quot; Said I, &quot;He's a Frenchman.&quot; &quot;Get out,&quot; replied Fleming, &quot;he's a 'bark.'&quot; &quot;Drinks he isn't,&quot; &quot;Done; drinks he is.&quot; &quot;Mike&quot; had not at that time spoken, but had noiselessly passed to and fro with a military step and a hand behind him. In reply to a call from Fleming he approached, and a &quot;Yes, yer honour,&quot; told me I should have to &quot;part.&quot; &quot;Mike&quot; was a trump, and I loved him from the start. For eleven years he had been in the service of a colonel at 1''l.'' a week, and three suit of clothes in the year. Hotel pay was not so good, and though he made &quot;a bit&quot; in the summer, the young ones were generally there when the bills were paid, if not when the work was on. &quot;Mike&quot; told us lots of things in what many consider the &quot;fascinating brogue.&quot; He informed us — and how our hearts beat at the news — that the island, which had a population of over 32,000, employed but six policemen. &quot;There's not a beggar in the land,&quot; said Mike, &quot;and but one Hansom cab, and,&quot; pointing to the window, &quot;be jabers, there it goes. &quot; So far so good, but after a walk round the son of Erin gave us a shock by saying, in reply to a query from Fleming, &quot;They've a way of their own on this island, and nearly ivery [sic] other man is a special constable.&quot; Fortunately we had just finished our repast, and, supporting each other, we tottered faintly to the door, partially regaining strength with a breath of the pure sea breeze. Beads of perspiration stood on the brow of the Pelican, whilst I was suffering from a &quot;trickling&quot; down the back of anything but a pleasing nature. Visions of a dungeon-cell under Castle Coronet flashed through my brain as a voice and a step were heard behind, and then I was somewhat reassured. &quot;When would you like a chair?&quot; fell softly on mine ear. Afraid that we had been thought of as a bit week on the leg, I turned to the speaker and asked what he meant. Explanations ensuing, we found the &quot;chair&quot; was a sort of Bath chair on wheels used throughout the island. Declining the offer, we took a stroll, and putting this and that together, came to the conclusion that our task was of a very delicate nature. The first thing was to interview the captain, who, in reply to a note, courteously waited upon us. He gave us the geography of the island and an introduction to Mr M., a friend, who we unfortunately found away from home. Taking a &quot;chair&quot; we then started on a voyage of exploration, making, as advised, our way to L'Ancresse [?] Common. Leaving St. Peter's Port and passing through St. Sampson's — the Vale — we came across acres of vineyards, in one instance extending nearly a mile along the road-side. Half an hour's drive brought us to the common mentioned, and dismounting we tried in vain to find a level piece of turf. Windmills abounded on all sides, and plodding on we reached L'Ancresse Bay, a lovely little spot, where the sea rippled softly over a bed of almost snow-white sand. Nowhere could be found a suitable spot whereon to pitch the &quot;magic square,&quot; and as the shades of night were falling fast we rattled back to the &quot;Royal.&quot; A wet night offered no inducement for further travel, and it was a case of early to bed and early to rise, the next day finding Fleming and myself at the door of Mr M. Acquainting this gentleman with the Nature of the business in hand he, like a good sportsman, fell into the scheme and at once volunteered to drive us to &quot;the best spot in the world.&quot; &quot;Bessie&quot; was harnessed to the village cart, and away we went at the rate of fully fourteen miles an hour in the direction of the Forest. I have sat behind many a good goer, but must for her size award the palm to Bessie a beauty barely twelve hands high. Throwing the reins in the bottom of the vehicle, Mr M. let the pony have her head, and the way she went uphill, down dale, and past other traps was truly astonishing. Great St. Peter's was passed, and lovely scenery encountered, the gardens being full of flowers, geraniums and roses blooming on all sides. An enjoyable drive of forty-five minutes found us at Boequaire [?] Bay, and directly in front stood the island of Lihou. As many of our readers may know, Lihou is a small hill of land that at low water can be reached on foot from Guernsey. As a fact, a paved roadway is laid in the bed of the ocean, across which both foot passengers and carts easily make their way. With the tide up communication is effected by boats, but this being low at the time we had the novel experience of   

Driving Through the Sea.   

Gaining the other side we &quot;explored&quot; and as the piece of land is only some three hundred yards by a hundred and fifty, we were not long about it. Two excellent patches of level turf were found, but just as we were making for the trap a gaunt, Robinson Crusoe-looking individual scaled one of the small stone walls and barred the way. Of course, he wanted to know our business, and, after a rather rough opening, quickly &quot;listened to reason.&quot; In the course of conversation we found that the population of the island was limited to three, the old &quot;josser&quot; and his two daughters, with a &quot;moke&quot; thrown in. Pointing a long skinny finger, which might have belonged to the Witch of Endor, towards the quadruped mentioned, the old man said, &quot;Us two keeps the family. He draws the seaweed and I burn it.&quot;  &quot;And why do you burn it?&quot; &quot;Well, to make [Col. 4c–5a] ashes, and from the ashes iodine is manufactured.&quot; &quot;And how much do yen earn a week by that?&quot; said  I. &quot;Oh, about 4''s.''; not 5''s.'' any week.&quot; Great Scott! and this was all the old man, his two daughters,  and the &quot;moke&quot; had to live on. Handing him a franc each, which nearly stupefied the old chappie, Fleming and myself returned to Mr M., who drove us rapidly back to St. Peter's Port. The next day we crossed to Sark and Jersey, but found nothing so suitable as the ground at Lihou, and fairly well satisfied with the result of our visit, a day later boarded the Lynx for the return journey.   

During our visit the weather had been of a fairly favourable description, but the voyage home was made under very depressing circumstances. Rain fell heavily and a thick mist hung over the water, which after going a few miles was alive with porpoise. Some of these animals came within a few yards of the Lynx, but the speed of the boat proved too good for the finney ones, who were eventually left behind. With a breeze springing up sail was hoisted, and dashing through a heavy sea in splendid form the Lynx made some sixteen knots in the hour. In this way half the journey was completed, when all of a sudden we were caught in a squall. In an instant the wind chopped round, and before you could say &quot;Jack Robinson&quot; the good vessel was turned broadside to. The order to take in sail was promptly obeyed, but the clatter on deck brought many an anxious passenger rapidly from below. A very heavy sea was now running, and though the Lynx kept her course bravely the decks were swept from stem to stern. A huge East India-man now hove in sight, and, as one of the men informed me, she had the audacity to cross our bows instead of passing under our stern. This breach of ocean etiquette ruffled the skipper of the Lynx, who chased the trader, and eventually made her alter her course. This little &quot;diversion&quot; caused much amusement amongst our passengers, and after a cold, rough passage we landed safely at Weymouth. Chilled to the bone, we made for the Victoria Hotel, where a chop and a glass of old October revived the almost stagnant circulation. London was reached about 9.30, and, tired and worn out, I was glad to lay my head on a pillow I had not seen for four days.   

From London to Ostend.

Unforseen [sic] difficulties having cropped up, the Channel Island scheme had to be abandoned; consequently we were once more called upon to leave our native land. This time Belgium was our destination, and having carefully studied the map of that country, we took our departure one fine starlight evening from Charing-cross by the 8.15 Continental express. In the previous week we had the luxury of a compartment to ourselves, but on the occasion under notice we were not so fortunate. Two individuals, one with a strong Midland accent, and the other a Frenchman speaking bad English, shared the carriage with us, and at such a rate did the former's tongue go directly he entered that I gave up reading as a bad job, with a sigh put away my unfinished book of the previous trip. Fleming fared better, and having a knack of ignoring all around him, once more tackled &quot;Blood Money,&quot; and almost read himself off to sleep.  In the meantime our friends in the opposite corner made &quot;hay,&quot; and I soon found out they were both in the lace trade. With a tongue that went a couple of hundred to the minute, the Midlander almost monopolised the conversation, from which I gleaned the two were bound for Calais, and thanked goodness we should part at Dover.

Reaching this port about ten we boarded the Ostend boat, which, as regards passengers, was very poorly patronised. Hardly a breath of wind was stirring, and with the sea calm as a mill-pond we had a pleasant passage over, and landed safely at Ostend soon after 3 a.m. A cup of coffee was hastily swallowed, and a half-hour's run in the Brussels express found us  [4, Col. 5b]

At Bruges.

Leaving the train, we had considerable difficulty in finding our way out of the station. For a moment we caught sight of a drowsy old porter, but the last carriage had not passed away from the platform ere he was lost to view. Seeing an opening on other side we made for it, only to have our passage barred by a half-dressed Belgian who, whilst displaying a total ignorance of English, Irish, and Scotch, directed us to return the way we came. For some time we tried in vain to find an outlet, and just as I had made up my mind to walk the platform until daylight Fleming found an iron gate ajar. Without loss of time we were in the street, and crossing the round, knocked loudly at the door of the nearest hotel. All was in darkness, and with no response to our fourth appeal for admittance, a return to the station or a &quot;tramp round&quot; appeared to be our only alternative. A slight noise inside followed by a shuffling of feet, again raised hope within us, and when, after some trouble, the door swung back and brough to view a shock-headed Belgian our relief was great. No time, I can assure you, was lost in getting between the sheets, and rising betimes we again went on a voyage of discovery. Fleming had a chum in the town, and to his house of call our steps were directed. Again disappointment was in store for our wanderers, and a feeling of disappointment came over me when we found Mr G. was away. There was, under the circumstances, nothing left but to wait, and Fleming, who puts on hair rapidly, decided to place himself under the care of the town barber, or, I should, properly speaking, my ''coiffeur''. For this piece of indiscretion he had to come away minus a slice of his lower lip, which, fortunately, was not large enough to seriously reduce his avordupois [sic?], and heaping blessings (?) [sic] on mankind generally, not in any way forgetting the &quot;Bruges Barber, we roamed over the old town, noted for its magnificent cathedral and many seats of learning. The hours sped on pleasantly enough, and in due time our friend was unearthed. Being pledged to secrecy I am not going from my word by giving full particulars of

The Battle Ground

placed at our disposal. This is in close proximity to the town — in fact, not half an hour's walk, and on reaching the spot we were shown a lovely piece of turn. A clump of trees on each side partly hid the house from view, and when I say that the ring was to be pitched on a lawn tennis ground, it will be seen that neither side could claim an advantage as regards standing ground, &amp;c., the lawn being as level as a billiard table. Though our efforts so far had been crowned with success, we did not get permission to use the &quot;pictch&quot; without much trouble. The matter had to be seriously considered, and Mr G. would not give a decided answer without waiting another day. Consequently our stay had to be extended and the following morning, after repeated promises that the affair should be conducted in an orderly manner with not a rough at the ring-side, we got the answer, &quot;Yes.&quot; &quot;But,&quot; said Mr G., &quot;if there is any row or crowd I can assure you you will have a journey for nothing; therefore if trouble is at all probable keep away and save your money and time.&quot; As long as faith was not broken by either side we saw no reason why the matter should not be settled in a proper manner, and making all the necessary arrangements and laying down plans of a most complete description, we returned to town well satisfied with the result of a visit to Bruges.  [4, Col. 5c]

Trouble about the Referee. 

On returning to town we received satisfactory reports of the work done by each of the contestants. Slavin, looked after by Josh Cosnett, the Birmingham heavy-weight, and his friend Mr Michael Taylor, of Australia, though rather light, was in good health. At one time he scaled only 12st 8lb, but a judicious spell of light work brought him back to 13st odd. Considering the importance of the business in hand, and its bearing on future events, Slavin had some  cause for anxiety. In addition to this he was fighting for 300''l.'' of his own money, and consequently wished to do himself justice in the international battle. 

The next question was the selection of someone capable and willing to fill the all-important post of referee. According to the articles it was left to the stakeholder to appoint the official named providing the representatives failed to agree, and in the end this course had to be adopted, solely owing to the objections lodged by Smith. The first gentleman suggested by Slavin's side was Mr B. J. Angle, whom Smith promptly objected to. Next a well-known owner of racehorses (a gentleman who adjudicated upon the knuckle fight between George Camp and &quot;Click&quot; Soles) was proposed with no better result. Mr Bat Murphy, strange to say, would not suit, and then in succession &quot;No&quot; was said to the names of  Messrs G. H. Vise, A. F. Bettinson, and [B.?] Bayley. All these gentlemen, it must be stated, were proposed by Slavin and objected to by Smith, who in the meantime had named a member of the Pelican Club, a gentleman of limited experience in these matters, and the landlord of the house from which he was training, Mr W. Riley, Newmarket. This brought about a deadlock, and the state of affairs getting wind, ugly rumours were soon afloat, and even money, and in many cases good odds, were betted that there would be &quot;no fight.&quot; All this while Slavin was chafing and fuming at the delay, whilst at the same time his  representative and Smith's were at work almost night and day endeavouring to bring about a settlement of the difficulty. For a long time it looked as though the matter would end in smoke, as after the services of the stakeholder had been called upon great difficulty was experienced in getting anyone to accept the office of referee. So late as Friday at noon nothing had been done, and it was not until the evening of that day that Mr Joe Vesey, consenting to stand, was nominated by the stakeholder. 

A Scare.

Though the battle-ground was only known to two persons, the country where the fight was to be decided being, in fact, or supposed to be, a perfect secret, on Friday morning a paragraph said to have been sent out by a news agency appeared in several papers, and created intense astonishment and consternation. This, set out in large type by those who we should have thought would have suppressed the account, stated that Smith on the previous evening had arrived in London from his training quarters, and even went so far as to name the street he slept in. That Smith's visit to town was extremely injudicious there is not the slightest doubt, but this was not the worst. The time of departure for the scene of action was given, with a description of the place selected. This publicity had the effect of keeping away many who bad expressed a wish to be present, and on Slavin's side no less than eight gentlemen, each holding a high position in the sporting world, declined to have anything further to do with the affair. Who divulged the secret is at present a mystery, but there is not the slightest doubt that the information so freely put forward not only opened the eyes of the authorities, but gave the &quot;mob,&quot; to a certain extent, a key to the [Col. 5c–6a] situation. Slavin received the news with no small amount of astonishment, and a couple of telegrams informing him that a batch of the &quot;boys&quot; had gone across to make things warm for him did not give peace to his troubled mind. Smith made his way out of England on Friday by the short sea route, Dover to Calais, which he reached in safety. Slavin, not so fortunate, was arrested on Satuday, as already reported, but, after being bound over to keep the peace, he got away the same evening.&lt;/blockquote&gt;

==== Performance of the Men. ====
&lt;blockquote&gt;Jem Smith

was born in Whitecross-street, in the parish of Cripple-gate, on January 21, 1863. He stands 5ft 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in high, and weighs in condition 12st 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;?&lt;/sub&gt;in; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; hips, 41in; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; calf, 16&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing, after one or two rough-and-tumbles with his mates he was taken in hand by Jack Knifton (&quot;81-tonner&quot;), and later by Mr F. Grimm, of the Central Club. Smith has never been defeated, though in his first engagement he did not shape like a coming Champion. His first success was in a competition at the Griffin, Shoreditch. Later he beat Snowey, of Holloway, and was then matched to box Harry Arnold, with gloves, for a 10''l.'' purse. Arnold, though a stone and a half lighter than Smith, stripped in good condition, having for some time previously been engaged in training Jack Massey for an important engagement. The affair was looked upon as a good thing for Smith, but it took him nearly an hour to polish off the Seven Dials' representative, though he had slightly the best of the encounter from start to finish. Subsequently the veteran Goode deposited a [sov?] on behalf of Smith to match him to fight Massey, but this was forfeited. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for 25''l.'' a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end twelve rounds. Smith also won Tom Symonds's All-comers' Competition on February 26[?], 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. At the close Smith was presented with a large oil painting of himself, executed by a prominent patron of the sport. Smith's next battle was with Jack Davis for 100''l.'' a-side and the Championship, on the turf, and came off on the borders of Surrey and Sussex, on Wednesday, December 17. In all six rounds were fought, occupying 7min 56sec, Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for 300''l.'' at Maisonne-Lafitte, near Paris, on Tuesday, February 16, 1886 [?]. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Jack Knifton, the &quot;81-tonner,&quot; as he is called. This proved a ''fiasco'', a two days' dreary tramp on the Continent being followed by a return to the old country. Knifton's people claimed that Smith had more than the specified number of followers, but there is little doubt that only one side wanted a fight, and that was not the &quot;81-tonner's&quot; lot. Meeting after meeting was held, Smith offering to fight anywhere the other liked, and after much wrangling arrangements were made to bring the affair off near Oxted [?]. Somehow of other the police got wind of it, and overtaking the Knifton party near Croydon rendered another postponement necessary. this time the arrangements were left in the hands of Knifton's side, with the result that the principals, seconds, and a dozen others were captured early one morning in a furniture van by the police before they had gone fifty yards as their journey. All were brought up at Thames Police-court, and after a remand bound over to keep the peace for twelve months. The memorial struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19, 1887 [?]. Jem's recent performance was a Ten-round Glove Contest with Jack Wannop, the Champion laying 250''l.'' to 200''l.,'' and winning easily, following which he was beaten in two rounds at the Pelican Club by Peter Jackson, the Coloured Champion, in a glove contest, for 1000''l.''  [4, Col. 6b] 

Frank P. Slavin. 

the Champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft 1in in his stockings, and in condition scales 13st 5lb or 13st 6lb. Slavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. He always had a liking for the gloves, and sparred a great deal when young, but did not enter into any match until the latter part of 1885 — that is, for money, though he won several competitions previously as an amateur. Slavin's first fight was with knuckles against M. Power for 50''l.'' a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with 2oz gloves for 200''l.'' a-side and the Championship of Queensland, winning in three rounds. This was in December, 1885 [?], and in the following year the Australian formed a combination and went touring, offering purses ranging from 10''l.'' to 20''l.'' to any man who stood up for four rounds. At this game Slavin had an easy time of it; some stood one round, and some two, but none of them four. Towards the end of the year he went to Sydney, and entered into an agreement with Larry Foley, who has a big place there. Here Slavin met Taylor, a pupil of Jackson, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big fellow named Bligh for 10''l.'' a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for 100''l.'' a-side. Dooley had a big reputation, but Slavin won in nine rounds. The subject of these remarks then had a little difficulty in finding an opponent, but eventually Mike Costello, an American, a good man, made his appearance, and the pair were matched for 100''l.'' a-side. They had a big battle in Sydney, lasting two hours and a half, and a draw was the result. Another match was made for 400''l.'', and 50''l.'' a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for 24''l.'' a-side to stop Costello in six rounds, but the latter is a very clever and tricky man, and lasted the time. Subsequently Frank journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for 100''l.'' a=side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for 250''l'' a-side and the Championship. Jackson's party wanted a match with gloves for a purse. They could not come to terms, though Slavin's backers offered Jackson 50''l.'' to sign articles for 500''l.'' Australian was now a bit bare, so in January last year the Australian made a trip to New Zealand. The first man put up was Snath, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the Champion. Terms could not be agreed upon until they had &quot;fiddled&quot; Slavin for nearly six weeks. The pair were then matched for 200''l.'' a-side, and fought with 2oz gloves at [Wangazul?], Laing failing to come up at the end of the fifth round. After this Slavin went back to Melbourne, hoping to get on a match with Jackson, and was surprised to find he had gone to America. Following this he met Jack Burke, but after Slavin had had all the best of it the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for 200''l.'' with small gloves, which ended in Mike being knocked out in 2min 40sec. He was very bad, and frightened all by not coming to for nearly half an hour. Slavin returned to Melbourne, and Burke having put several challenges in the paper, our visitor offered to stop him in six rounds; 200''l.'' a-side was put up and articles signed for 4oz gloves, but on the night Burke's party threw down regular pillow-cases. Of course this did not suit the other side, and there was a fine old wrangle, lasting two hours and a half. Eventually Mr Joe Thompson, who was referee, declared all bets off, and the two fought 6oz gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for 200''l.'', with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. This is considered to be one of Slavin's best performances. About this time Woolf Bendoff, of Whitechapel, was in Melbourne, and finding plenty of backers, they put up a forfeit of 100''l.'' for the Londoner to box Slavin, for 500''l.'' a-side. After seeing the latter polish off Burke, however, Bendoff's friends would go no further with the atch, and Slavin accordingly pocketed the &quot;century.&quot; An easy victory over Bill Buck in two rounds followed, and then Slavin, on October 17 made his first appearance in England at Astley's Theatre, where he beat &quot;Chesterfield&quot; Goode in five rounds in a match of twelve rounds, for 400''l.''  [4, Col. 6c] &lt;/blockquote&gt;

==== THE PELICAN CLUB AND THE FIGHT. ====
&lt;blockquote&gt;To the Editor of the Sportsman. 

Sir, — As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it. — Yours truly,  

Cecil Raleigh, Hon. Sec. 

Pelican Club, 21, Denman-street, W., Dec. 23. 

__________ 

We hear on good authority that it is the intention of the members of the Pelican Club to present Slavin, the Australian Champion, with a belt, in recognition of the manly exhibition he gave yesterday at Bruges. &lt;/blockquote&gt;

==== THE CHAMPIONSHIP OF THE WORLD. ====
&lt;blockquote&gt;Frank Slavin to John L. Sullivan. 

Frank Slavin, the Champion of Australia, is prepared to meet John L. Sullivan, under London Prize Ring Rules, for 500''l.'' or 1000''l.'' a-side and the Championship of the World. Slavin has lodged with us a sum of 100''l.'', and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr R. K. Fox, of New York, and the money lodged with him or the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; will lead to a match.&lt;ref name=&quot;:1&quot; /&gt; &lt;/blockquote&gt;

=== Dundee ''Evening Telegraph'' ===
This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:&lt;blockquote&gt;There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.&lt;ref&gt;&quot;London Gives the Colonist a Great Reception.&quot; ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&amp;page=3&amp;article=052&amp;stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.&lt;blockquote&gt;The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; 

Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot;Police!&quot; Smith was hurried from the ring; but Slavin and his friends refused to budge, and, after &quot;Time&quot; had been called, they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to Margate. Smith and his friends were left behind at Bruges.

Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec.

Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other.

Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.)

Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him.

Round 5. — A light round, Smith going down before a light blow. Time, 11sec.

Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead.

Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners.

Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground.

Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min.

Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together.

Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander.

Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec.

Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin.

Round 14 and last. — Just a second before &quot;Time&quot; was called, Slavin, remarking on the treatment he had received, said, &quot;Are you Englishmen, boys?&quot; This was received with derisive shouts from Smith's party, and then, in an instant, a cry of &quot;Police&quot; was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, &quot;Why don't you give me fair play?&quot; Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, &quot;I shall refuse to act any longer. This man&quot; — pointing  to Slavin — &quot;can't have fair play. I declare it a draw.&quot; Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there.

Here the fight came to an end, the referee exclaiming, &quot;I shall refuse to act any longer. I shall declare it a draw.&quot; There was considerable surprise among Slavin and his supporters, and the Australian said, &quot;I came 16,000 miles to win, and why don't you give a man a chance?&quot; After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine.

Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— &quot;As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it.&quot; 

Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. 

That Slavin has now a right to the title (empty though it be) of &quot;champion&quot; is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— &quot;Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. 

Arrival of Slavin at Margate. 

Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, &quot;See the conquering hero comes.&quot; Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, &quot;Well, Slavin, how have you fared?&quot; &quot;Very badly,&quot; responded the Australian. &quot;As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders.&quot; — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. 

The Pelican Club and the Fight. 

A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— &quot;In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business.&quot; 

It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. 

The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— 

&quot;In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, &quot;F. P. SLAVIN. 

&quot;Nayland Rock Hotel, Margate.&quot; 

Mr. Fleming's Statement. 

Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. 

The Preliminary Arrangements. 

The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the &quot;lost art&quot; was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the &quot;old style.&quot; Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. 

There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. 

Performances of the Men. 

Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. 

Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; hips, 41in.; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; calf, 16&lt;sup&gt;3&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in.; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton (&quot;eighty-one tonner&quot;) and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the &quot;eighty-one-tonner&quot; as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.&lt;ref&gt;&quot;The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring.&quot; ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&amp;page=1&amp;article=006&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== The ''People'' ===

On Sunday, 29 December 1899, the ''People''&lt;blockquote&gt;Disgraceful Scenes. Expressions of indignation.

The prize fight between Jem Smith, who poses as the fistic &quot;champion of England,&quot; and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a &quot;strong corner.&quot; The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that &quot;Smith did not lose&quot; proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper.

The Fight.

The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot; Police!&quot; Smith was hurried from the ring, but Slavin and his friends refused to budge, and after &quot;Time&quot; had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of &quot;Foul,&quot; the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate.

Slavin's Opinion of his Treatment.

The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there.

Indignation in London.

Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee.

The Pelican Club Meeting.

At the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of &quot;the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.[&quot;] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed.

Slavin Challenges J. L. Sullivan.

Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&amp;page=16&amp;article=150&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Sportsman'' ===
On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the &quot;testimonial&quot; fund for Slavin as well as some letters to the editor about the Slavin-Smith match:&lt;blockquote&gt;
A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). 

The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris.

The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.'']

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Lord James Douglas  2 2 0
* Hon. Michael Sandys  2 2 0
* Fred Kotchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Proprietors of The Sportsman  2 2 0
* Mr. J. Purdey  2 2 0
* Mr. E. W. Bayly  2 2 0
* Messrs H. A. Domby and Son  2 2 0
* &quot;Caption Coe&quot;  2 2 0
* Mr C. W. Bulpett  2 2 0
* Mr P. W. Carr  2 2 0
* Mr W. Morley Pegge  2 2 0
* W. D. S.  2 2 0
* J. E. S.  2 2 0
* Mr W. H. Grenfel  2 2 0
* Mr Quartermaine East  2 2 0
* Mr H. H. Hungerford  2 2 0
* Mr Noel Fenwick  2 2 0
* Sir John Leslie, Bart.  2 2 0
* Mr T. Brinkman  2 0 0
* Mr Calder  1 1 0
* Mr H. W. Hill  1 1 0
* Mr John Young  1 1 0
* Mr D. A. Neilson  1 1 0
* Mr W. Neilson  1 1 0
* Mr Julius Simon  1 1 0
* Mr J. Edge  1 1 0
* Mr H. Marshall  1 1 0
* Mr F. Aynsley  1 1 0
* Col. Hon. Wellesley  1 1 0
* Mr Charles Hawtrey  1 1 0
* Mr W. F. Ashton  1 1 0
* Mr Barratt  1 1 0
* Old Durham Ox  1 0 0
* Sir John Willoughby  1 0 0
* A friend, per C. H. A.  0 10 6
* Mr Ben Giles  0 10 6
* Mr Tom Vale  0 10 6
* [E.?] W.  0 10 6
* Officer  0 10 0
* Per Mr Ike White, Exchange Club:
** S. W.  2 2 0
** Mr G. B. Knight  1 1 0
** Mr A. Powles  1 1 0
** Mr S. Blandford  1 1 0
** T. F. W. C.  1 1 0
** Mr Leo Harward  1 1 0
** Mr J. Harward  1 1 0
** Mr H. Harward  1 1 0
** Mr [Monteagle?] [Moutcagie??]  1 1 0
** Mr C. Orate  0 10 6
** W. G. P.  0 10 6
** H. R. T  0 10 6
** Mr Josiah Smith  0 10 6
** W. H. R.  0 10 6
** Mr R. Wilson  0 5 0
* Per Mr M. J. Harlow, Margate:
** Mr A. H. [Cosi? Coni?]  2 2 0
** Mr J. Thurston  1 1 0
** Mr J. A. Fagg  1 1 0
** Mr F. [Miles?]  1 1 0
** Mr Adolphe [Drineqbeir?]  1 1 0
** Mr Ernest [Drineqbier?]  1 1 0
** Mr Lacy Brown  1 1 0
** Mr Algy Brown  1 1 0
** Mr M. J. Harlow  1 1 0
** An Admirer  1 1 0
** Mr J. W. Robinson  1 1 0
** Mr W. Searles  1 1 0
** Mr J. Smith  1 1 0
** Mr J. Hume  1 1 0
** Mr J. Painter  1 1 0
** A Scotchman  1 1 0
** Mr J. Mundell  0 10 6
** Mr Bob Redman  0 10 6
** Mr G. A. Body  0 10 6
** Mr J. Phipps  0 10 6
** Mr J. Doughty  0 10 6
** Old Fisherman  0 10 6
** Mr Joe Knott  0 10 6
* Per Mr B. J. Angle
** Some mebers of the Stock Exchange  125 0 0
** Hr Henry J. Turner  2 2 0
* Per Mr W. J. Innes:
** Mr Sidney Fry  2 2 0
** Mr W. J. I.  2 2 0
** Mr W. Schlesinger  2 2 0
** Mr Joseph Thompson  2 2 0
** Mr Barney Thompson  2 2 0
** Mr L. Meyers  2 2 0
** Mr W. Shoe  2 2 0
** Mr George Cooper  2 2 0
** Mr J. T. Crossley  2 2 0
** Mr Andy Anderson  2 2 0
** Mr G. M. T.  2 2 0
** Mr F. R.  2 2 0
** Mr W. Bertram  1 1 0
** Mr J. W. Hammon  1 1 0
** Mr Alec Harris  1 1 0
** Mr John Isaacs  1 1 0
** Mr Edward [Dance? Dasce?]  1 1 0
** Mr George Blanchett  1 1 0
** Mr F. [Leleu? Lolou?]  1 1 0
** Mr W. J. R.  1 1 0
** Mr H. C.  1 1 0
** Mr A. A. M.  1 1 0
** Mr D. W. Fowler  0 10 6
* Per Mr F. H. Gale:
** Mr R. Moore  2 2 0
** Mr A. H. Goode  2 2 0
** Mr [B.? E.? R.?] J. G. Goode  2 2 0
** Mr F. H. Gale  2 2 0
** Mr H. Ulph  1 1 0
** Mr J. Hooper  1 1 0
** Mr C. Kittall  1 1 0
** Mr C. Moore  1 1 0
* Per Mr W. Riley:
** Mr Brand (Wiltshire)  2 2 0
** Mr Charles Godfrey  2 2 0
** Mrs Lane (Britannia)  2 2 0
** Mr Sam Adams  2 2 0
** B. and M.  0 7 6
* Per Mr A. Cockburn:
** Mr Richardson  2 2 0
** Mr A. Cox  1 1 0
** Mr Hy. Brackenbury  1 1 0

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and &quot;knuckle-dustered&quot; him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, &quot;Come and have a round.&quot; The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called &quot;seconds&quot; should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of &quot;Police&quot; was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska &quot;tough,&quot; as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how &quot;seconds&quot; — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a &quot;duster&quot; might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &amp;c.,

A Lover of Boxing.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &amp;c.,

Leicester, December 28. J. Barrett.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded.

I cannot, however, agree with those who hope &quot;that we have seen the last of prize-fighting,&quot; and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &amp;c.,

J. W. B. C., Junior Carlton Club, December 27.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself &quot;England's Champion.&quot; I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon &quot;British fair-play and courage.&quot; — Yours, a lover of it, Sydney Galvayne. 

York, December 26. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, 

Birmingham, December 27. Tom B. Sharp. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &amp;c., 

I. White.  

Exchange Club, Queen-street, Cheapside, Dec. 28.&lt;ref&gt;&quot;The Ring. The Slavin Testimonial.&quot; ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&amp;page=4&amp;article=020&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; &lt;/blockquote&gt;
On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:&lt;blockquote&gt;JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.
{{paragraph break}}

J&lt;small&gt;OHN&lt;/small&gt; L S&lt;small&gt;ULLIVAN&lt;/small&gt; has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me.&quot;&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:&lt;blockquote&gt;THE SLAVIN TESTIMONIAL
{{paragraph break}}
The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office.&lt;/blockquote&gt;
{| class=&quot;wikitable&quot;
!
!£
!s.
!d.
|-
|Amount already acknowledged
|407
|3
|6
|-
|Mr E.Tabernacle
|1
|1
|1
|-
|H. S.
|1
|1
|0
|}
&lt;blockquote&gt;The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Huddersfield Daily Chronicle'' ===
On 10 January 1890:&lt;blockquote&gt;
Mitchell and Lord Mandeville.

As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of &quot;sports&quot; at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by &quot;Pony&quot; Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &amp;c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. &quot; [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and &quot;went for&quot; Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and &quot;Pony&quot; Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was &quot;something up,&quot; and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&amp;page=4&amp;article=071&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;
After the Pelican Club meetings, Slavin comments:&lt;blockquote&gt;On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English &quot;champion,&quot; made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.&lt;ref&gt;&quot;Slavin at the Royal Acquarium.&quot; ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&amp;page=3&amp;article=042&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:&lt;blockquote&gt;Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. &quot;This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of &quot;heelers&quot; always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me.&quot;&lt;ref&gt;&quot;The Recent Prize Fight.&quot; ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&amp;page=6&amp;article=053&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Cork Constitution'' ===
About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:&lt;blockquote&gt;This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s &quot;lefts.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Colonial Standard and Jamaica Despatch'' ===
This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:&lt;blockquote&gt;The disgraceful affair, called by courtesy and [sic] &quot;International Prize Fight,&quot; which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The &quot;fight&quot; lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn [??, s/b Royston?], Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.&lt;ref&gt;&quot;London.&quot; ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&amp;page=2&amp;article=019&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;
== Notes and Questions ==
# 

==References==
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International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium.

==Overview==
On 23 December 1889 a championship boxing match in Bruges, Belgium, ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin or getting him near the ropes and beat him up, using fists, sticks, &quot;knuckle-dusters,&quot; &quot;life preservers&quot; and boots. The referee ruled the match a draw to prevent Slavin from being killed and then fled. The ''Cork Constitution'' called it &quot;the Smith-Slavin piece of brutality.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;

A knuckle-duster was &quot;A metal instrument made to cover the knuckles,&quot;&lt;ref&gt;“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.&lt;/ref&gt; like brass knuckles, and a life preserver was &quot;A stick or truncheon weighted at one end&quot; with lead.&lt;ref&gt;“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.&lt;/ref&gt;

Running throughout the longest article are two ideas:

# that what rules to use is a choice to be made: the Prize Ring rules, the Queensberry rules, some rules adopted and numbered
# that the men choosing the grounds for the fight, the gentleman who ultimately hosted the fight and the people who attended the fight do not want to attract the attention of law enforcement of whatever kind

One meeting of the boxing-oriented London Pelican Club was held almost immediately after the fight. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below).

Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England.

Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made here to offer complete coverage. Specialized sports coverage is privileged.

==Logistics==
* 23 December 1889, Monday, 8:00–9:00 a.m., actual fighting commencing at 8:35
* The &quot;private grounds of a gentleman's residence near Bruges, Belgium&quot;&lt;ref name=&quot;:1&quot;&gt;&quot;The Great Fight. Smith &amp; Slavin for £1000. Disorderly Proceedings. Brutal Attack on Slavin. Smith Knocked Down Repeatedly. He Leaves the Ring. A Draw Declared after 14 Rounds.&quot; ''The Sportsman'' 24 December 1889, Tuesday: 3 [of 4], Col. 1a – 4, Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&amp;page=3&amp;article=057&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; (3, Col. 1a). The gentleman was Mr. G., or 
* Stakes: £1,000
==Related Events==
* [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]]
* the &quot;[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]&quot; 
==Who Was Present==
# Jem Smith, British boxer
## Jack Harper, Smith's second
## Jack Baldock, Smith's second
## Mr J. Fleming, umpired for Smith, &quot;the manager of the Pelican boxing,&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 4a) travelled with G. T. Dunning to find the grounds where the fight would take place, Fleming representing Smith and Dunning representing Slavin
# Frank Slavin, Australian boxer
## Josh Cosnett, Slavin's second
## Bob Kirby, Slavin's second
## Mick Taylor, &quot;superintended the arrangements in his [Slavin's] corner&quot;
## Mr W. Riley, umpired for Slavin
# &quot;the company round the ring
## Lord Mandeville — [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]]
## Lord Royston
## Hon. M. Sandys
## Mr Abington — Mr. G. Abington Baird, Smith's manager? and a horseman (see description in Anthology)
## Mr F. Kotchie
## Mr J. Kilby — Jack Kilby, &quot;a real sportsman&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## Mr C. Parsons
## Mr Ike White
## Mr C. Morton
## Mr Harris
## Mr E. Bayley — Teddy Bayley
## Mr John Lewis (Australia) — Jack Lewis, Slavin's manager
## Mr J. Vesey — Joe Vesey, more often, or Vezey, referee
## Mr F. Hinde — Frank Hinde
## Mr M. Taylor
## Mr J. Calder — timekeeper
## &quot;Pop&quot; Joy
## R. Habbijam — Bob Habbijam
## G. (&quot;Barney&quot;) Sheppard — &quot;superintended the pitching of the ring&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3b)
## Jem Carney
## Tom Smith
## Mangie [?] Hyams
## Matthews
## Summers
## Boulton
## Mr G. T. Dunning — Tom Dunning, worked at ''The Sportsman'' and served as treasurer for the subscription to collect a cash prize for Slavin; Dunning and Fleming traveled to the Channel Islands and the continent to find a ground for the fight to take place in, and Dunning probably wrote the account of it published in ''The Sportsman''.&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 4a)
## &quot;several others, including two or three reporters&quot; — &quot;three scribes with pockets bulging from a surplusage of note-books dropped in&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1c)
## The writer of the big 24 December article, who reports for ''The Sportsman'' and who is not the &quot;well-known sporting journalist&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 1b) who had heard that Slavin had been arrested in Margate
# &quot;an English resident in Bruges&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 3a)
# &quot;Pop&quot; Hinde
# C. Horton
# &quot;a mob of Birmingham men&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
# Lord Rosslyn?, probably typo or mistake for Lord Royston
# &quot;The Squire&quot;
#
==Entertainment==

==Anthology==

=== The Sportsman ===
The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 and most of page 4 (nearly 13 full columns) entirely to this story. This reporter's writing style is very ornate:

* &quot;Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping&quot; (3, Col. 1c)
* &quot;For a time it [the sea] was a bit rough, and one of the good sailors commenced quoting Byron. 'O'er the glad waters of the dark blue sea,' sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2a)
* &quot;Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2b)
* &quot;While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)
* &quot;He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training.&quot;&lt;ref name=&quot;:1&quot; /&gt; (3, Col. 2c)

Besides the street-fighting terms (knuckle-dusters, life preservers) above, he also uses slang and local vocabulary like sabots (wooden clogs worn by working-class Belgians), Brums (people from Birmingham [?])

This is, then, a very long article, reproduced here in full except where the text is illegible, the very large block quote broken up by level-2 heads for navigation:&lt;blockquote&gt;The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below.&lt;/blockquote&gt;

==== THE JOURNEY TO THE CONTINENT. ====
&lt;blockquote&gt;(By our special reporter.)

[?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of &quot;globe-trotting,&quot; as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came]

A Rude Shock.

[Arriving??] at the office of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that &quot;something had happened.&quot; His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. &quot;What do you think of this?&quot; asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. &quot;Stoppage, but think can get bail. — Slavin.&quot; Here was a pretty &quot;go,&quot; the mystic words being translated meaning that the Australian had been &quot;pinched&quot; at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it

A May Morning.

Long before '''the earliest purveyor of the lacteal fluid''' arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of &quot;London peculiar&quot; had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking &quot;all broke up,&quot; put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a '''surplusage of note-books''' dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, &quot;Pop,&quot; and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey [sic], who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross &quot;The Squire,&quot; Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''.

Down to Dover.

It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly &quot;gave the game away,&quot; and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as &quot;dirty&quot; weather in store.

On the Briny.

The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and &quot;once on board the lugger&quot; my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the &quot;Princess&quot; swung off from the pier the two &quot;Bobs,&quot; Kirby and Habbijam, betook themselves to the &quot;rail&quot; in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. &quot;O'er the glad waters of the dark blue sea,&quot; sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having &quot;sought the seclusion which the cabin grants,&quot; and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in &quot;Pop,&quot; and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. [3, Col. 2a–2b] 

Ostend to Bruges.

While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded &quot;foreigner&quot; who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, &quot;Well, when is the 'mill' coming off?&quot; In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any &quot;mills.&quot; As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring '''that useful saponaceous compound''' with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity. [3, Col. 2b–2c]

Pugilism at the Piano.

While '''awaiting the refection''' Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys soon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The &quot;pelting of the pitiless storm&quot; continued the greater part of the night. 

Arrival of Jem Smith. 

The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arrangements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or &quot;the manger&quot; as &quot;Pop&quot; expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery &quot;Good morning, gentlemen,&quot; Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. 

The March to the Battlefield. 

Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out again. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work [3, Col. 3a–3b] 

Building the Ring. 

Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. 

The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, &quot;Pop&quot; Joy, R. Habbijam, G. (&quot;Barney&quot;) Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that 

The Dreaded Birmingham Division 

had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the &quot;boys&quot; worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, &quot;I hope all those present will keep quiet and help me all you can,&quot; at which brief speech there was a chorus of &quot;hear, hear,&quot; especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. 

The Call of &quot;Time&quot; 

At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, 

Smith was Knocked Down 

By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their &quot;Champion&quot; did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the &quot;Champion&quot; wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to &quot;be a man.&quot; Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of &quot;Here they are&quot; was raised, and 

A Stampede Ensued 

from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, &quot;There they are, there they are.&quot; By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, &quot;Are you Englishmen? I am an Englishman, and only want fair play.&quot; The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a &quot;duster,&quot; the referee saying, &quot;I saw that.&quot; As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, &quot;I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play.&quot; The referee concluded by giving the decision, &quot;It's a draw.&quot; The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting &quot;There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance.&quot; There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. 

There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended 

Making All Speed Homewards. 

The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. 

Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. &lt;/blockquote&gt;

==== THE FIGHT. ====
&lt;blockquote&gt;First Round. 

Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. 

Second Round. 

No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, &quot;First blood,&quot; a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On &quot;Time&quot; being called the pair were quick to scratch, that is to say, Slavin walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec.

Third Round.

Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking back to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec.

Fourth Round.

Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, 1min 15sec.

Fifth Round.

&quot;Don't go into his corner,&quot; urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite corner, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds.

Sixth Round.

Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec.

Seventh Round.

In vain did Slavin's friends implore him not to go into Smith's very &quot;warm corner.&quot; It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion was something terrible, and the ring looked very much like being invaded. The ropes in Smith's corner were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. 

Eighth Round.

Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth. Time of round, 1min 30sec.

Ninth Round.

This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a]

Tenth Round.

The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. 

Eleventh Round. 

Neither was quite so quick to time, and Harper, noticing that, said, &quot;This is not a two-round job, Patsey.&quot; Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to &quot;Come on and have a round.&quot; This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. 

Twelfth Round. 

A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. 

Thirteenth Round. 

Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, &quot;Oh!&quot; Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of &quot;Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a &quot;blue.&quot; The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words &quot;fight on&quot; came he and his friends were very delighted. 

Fourteenth and Last Round. 

When &quot;Time&quot; was called Slavin folded his arms and waited for Smith to &quot;come on,&quot; but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. 

No sooner was this round over than the referee called out. &quot;I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw.&quot; Then, going up to Slavin he said, &quot;I will not stay to see you killed.&quot; The Antipodean stood up and exclaimed, &quot;If you are Englishmen why don't you give me fair play? I have not had fair play.&quot; 

After this came a stampede. 

Remarks. 

Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as &quot;fights in the old style.&quot; That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the &quot;upper crust.&quot; Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being &quot;bashed&quot; on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with &quot;dusters&quot; and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a &quot;glover.&quot; With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a] that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or &quot;bottom&quot; of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, &quot;he was just getting warm.&quot; Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect.  

For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved.

From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play.

Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow. Bearing on this subject, Rule 26 says:— &quot;That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle.&quot; Rule 24 is also plain enough in all conscience, &quot;That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the &quot;boys&quot; and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but &quot;Kim&quot; soon showed he could hold his own with the &quot;boy,&quot; who beat a hasty retreat at the &quot;sight&quot; of six-inch &quot;Jack.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 1a–1b)

We now take our leave of this unsavoury affair, which will probably be the last of its kind — &quot;a Championship knuckle fight.&quot; The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner &quot;merry mills in the old style&quot; die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office about three o'clock.

Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of &quot;dusters&quot; the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye.

It may be stated that not a single bet was made at the ring side. [4, Col. 1b]&lt;/blockquote&gt;

==== SLAVIN'S OPINION OF HIS TREATENT. ====
&lt;blockquote&gt;His Future Plans. 

On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. 

Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. 

With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. 

Much sympathy was expressed on all sides, and in the course of conversation with a S&lt;small&gt;PORTSMAN&lt;/small&gt; representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] 

Did you expect any trouble at the ring side? 

Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. 

Do you feel any effects of the fighting? 

Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. 

Why did you persist in going to his corner? 

Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. 

Of course you heard a mob was got together for Smith? 

Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. 

What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. [4, Col. 1c] 

I suppose you have a poor opinion of English fair play? 

Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. 

What are your plans for the future? 

Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. 

With the gloves? 

Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. 

You prefer the old style to the new? 

Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. 

Shall you give any exhibitions? 

Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. 

You put on weight the last week? 

Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. 

What do you think of our seconds? 

Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. 

When do you propose visiting London? 

Slavin: I'll come up to-morrow and call at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office about three. 

A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. &lt;/blockquote&gt;

==== A Testimonial for Slavin. ====
&lt;blockquote&gt;So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.'' The maximum donation was 2 guineas, which would be 2£ 2''s.''] 

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Hon. Michael Sandys  2 2 0
* Mr Fred Ketchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Mr J. Calder  2 2 0

Any contributions sent to the hon. treasurer will be duly acknowledged in T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;, which in a couple of days will contain the names of a committee now in course of formation. 

In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; 

Sir, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  &quot;Champion of England&quot; to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. 

If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C.&lt;br&gt;
Royal Thames Yacht Club, December 23. 

[We refer our correspondent to the paragraph above. — Ed. T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. ] [sic bracketed interpolation]  [4, Col. 2b] &lt;/blockquote&gt;

==== Eyewitnesses Interviewed. ====
&lt;blockquote&gt;A representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known &quot;sports&quot; (including more then one scion of a noble house) for a few minutes' chat. 

Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which did not number more than forty, being particularly noticeable. 

&quot;How did the combatants' seconds behave?&quot; — &quot;Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes.&quot; 

&quot;What form was the Englishman displaying?&quot; — &quot;From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds. Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy.&quot; 

&quot;How do you account for such a noisy corner at the back of Smith?&quot; — &quot;Although they did not number more than eight, they were some of the worst characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, inasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, &quot;framed&quot; [sic double quotes] up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his party.&quot; 

Are these men known at all?&quot; — &quot;Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period.&quot; 

&quot;Was not this sort of trouble expected?&quot; — &quot;It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated.&quot; 

In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. 

&quot;How did the crowd behave towards the finish?&quot; — &quot;There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious.&quot; 

&quot;Is it true that attempts were made to tamper with the water used by Slavin's seconds?&quot; — &quot;Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail.&quot; 

&quot;Did Smith deal out much punishment?&quot; — &quot;His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought for two hours more. He could have beaten Smith and both his seconds into the bargain, given fair play.&quot; 

&quot;Was Slavin injured during the fight?&quot; — &quot;No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed.&quot; 

&quot;As to the referee's decision?&quot; — &quot;That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened.&quot; 

&quot;What becamiof Smith after the fight.&quot; — &quot;He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does.&quot; 

&quot;How did the Cornstalk behave in the ring?&quot; — &quot;He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion.&quot; 

Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a &quot;mill,&quot; was very severe in his comments on the gang who were so lost to all sense of decency and fair play. 

Mr White said: — &quot;The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He had to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful.&quot;  [4, Col. 2c] 

&quot;Do you think it was for the best that the contest was stopped as it was?&quot; 

&quot;Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]&quot; 

&quot;What do you think of Slavin's abilities?&quot; 

&quot;He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt.&quot; 

On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. &quot;He is coming to town to-morrow,&quot; said the speaker, &quot;and shall have a great reception. Some good sportsmen have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it.&quot; 

&quot;What seemed the general opinion on the ground?&quot; 

&quot;Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took.&quot; &lt;/blockquote&gt;

==== Reception of the News in Birmingham. ====
&lt;blockquote&gt;Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a] and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. 

How the Match Was Made 

On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— &quot;He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, was unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers.&quot; [sic: quotation begins at the top of the paragraph] 

Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; and signed:   

Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', for the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally.   

In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names. [4, Col. 3c] &lt;br&gt;
Frank Hinde (for Frank Slavin). &lt;br&gt;
John Fleming (for James Smith). &lt;br&gt;
Witness for Smith, F. Montague. &lt;br&gt;
Witness for Stavin, John Lewes. &lt;br&gt;

Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The &quot;Cornstalk,&quot; though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not stop in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds.   

Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell.   

Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[.] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice.
&lt;/blockquote&gt;

==== The Men in Training ====
&lt;blockquote&gt;Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp.   

Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound, Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson. [3, Col. 4a]   &lt;/blockquote&gt;

==== Finding the Battle-Ground. ====
&lt;blockquote&gt;(By one of the Explorers.)   

At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class &quot;smoker&quot; attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed that a &quot;choppy sea&quot; was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. &quot;How confoundedly annoying,&quot; he said. &quot;This is the sixth time I have been across during the year, and not once has it been calm.&quot; &quot;Better have a sleeping berth, sir,&quot; said a sprightly youth at the top of the stairs. &quot;Sleeping berth be hanged. I'll remain on deck,&quot; said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit &quot;peckish&quot; and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the &quot;magging&quot; [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, &quot;struck ile [?]&quot; with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The &quot;elderly party&quot; had a voice greatly resembling the dividing of tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. Just as I thought of giving up sleep for a bad job a gale sprang up. Though the Geselle is a record breaker she soon felt the effects of the breeze and began to roll heavily. The increase in the motion of the steamer was followed by a decrease in the conversation around me, and by degrees the punster went from had to worse. The &quot;[hussey??]&quot; [one?] also dropped into a walk, and after monosyllables had been exchanged the two came to a dead stop. Offering up a prayer of thanksgiving for this I closed my eyes again, and being a good sailor was quickly sleeping the peace of, I hope, the just. The concussion of the Gazelle against the Guernsey Pier, accompanied by a prod from Fleming's stick, at length aroused me.  [4, Col. 4b]   

Among the Channel Islands.   

The time by &quot;Joe Vickers&quot; was 7 a.m., and half-awake I took to my boots. &quot;Need not lace them; we shall be there in a minute,&quot; said Fleming, and in a drowsy, submissive, take-me-anywhere-if-only-to-bed sort of way I followed him on deck. A Babel of voices and a clutch at my bag and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a &quot;[sheenie?]&quot; (goodness knows what he meant, though), made us welcome, and agitating the communicator he handed us over to the care of a dark-eyed chambermaid. I forget her name, and as Fleming is not here I cannot give it. I will say no more than that after a cup of tea diluted with delicious cream, we took a well-earned and much needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the &quot;holiday maker&quot; and the &quot;honeymooner,&quot; is, as everybody knows, or should know, situate in the English Channel near the coasts of Brittany and Normandy. It is 68 [?] miles from Weymouth, 7 from Sark, 19 from Alderney [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in &quot;Mike.&quot; Said I, &quot;He's a Frenchman.&quot; &quot;Get out,&quot; replied Fleming, &quot;he's a 'bark.'&quot; &quot;Drinks he isn't,&quot; &quot;Done; drinks he is.&quot; &quot;Mike&quot; had not at that time spoken, but had noiselessly passed to and fro with a military step and a hand behind him. In reply to a call from Fleming he approached, and a &quot;Yes, yer honour,&quot; told me I should have to &quot;part.&quot; &quot;Mike&quot; was a trump, and I loved him from the start. For eleven years he had been in the service of a colonel at 1''l.'' a week, and three suit of clothes in the year. Hotel pay was not so good, and though he made &quot;a bit&quot; in the summer, the young ones were generally there when the bills were paid, if not when the work was on. &quot;Mike&quot; told us lots of things in what many consider the &quot;fascinating brogue.&quot; He informed us — and how our hearts beat at the news — that the island, which had a population of over 32,000, employed but six policemen. &quot;There's not a beggar in the land,&quot; said Mike, &quot;and but one Hansom cab, and,&quot; pointing to the window, &quot;be jabers, there it goes. &quot; So far so good, but after a walk round the son of Erin gave us a shock by saying, in reply to a query from Fleming, &quot;They've a way of their own on this island, and nearly ivery [sic] other man is a special constable.&quot; Fortunately we had just finished our repast, and, supporting each other, we tottered faintly to the door, partially regaining strength with a breath of the pure sea breeze. Beads of perspiration stood on the brow of the Pelican, whilst I was suffering from a &quot;trickling&quot; down the back of anything but a pleasing nature. Visions of a dungeon-cell under Castle Coronet flashed through my brain as a voice and a step were heard behind, and then I was somewhat reassured. &quot;When would you like a chair?&quot; fell softly on mine ear. Afraid that we had been thought of as a bit week on the leg, I turned to the speaker and asked what he meant. Explanations ensuing, we found the &quot;chair&quot; was a sort of Bath chair on wheels used throughout the island. Declining the offer, we took a stroll, and putting this and that together, came to the conclusion that our task was of a very delicate nature. The first thing was to interview the captain, who, in reply to a note, courteously waited upon us. He gave us the geography of the island and an introduction to Mr M., a friend, who we unfortunately found away from home. Taking a &quot;chair&quot; we then started on a voyage of exploration, making, as advised, our way to L'Ancresse [?] Common. Leaving St. Peter's Port and passing through St. Sampson's — the Vale — we came across acres of vineyards, in one instance extending nearly a mile along the road-side. Half an hour's drive brought us to the common mentioned, and dismounting we tried in vain to find a level piece of turf. Windmills abounded on all sides, and plodding on we reached L'Ancresse Bay, a lovely little spot, where the sea rippled softly over a bed of almost snow-white sand. Nowhere could be found a suitable spot whereon to pitch the &quot;magic square,&quot; and as the shades of night were falling fast we rattled back to the &quot;Royal.&quot; A wet night offered no inducement for further travel, and it was a case of early to bed and early to rise, the next day finding Fleming and myself at the door of Mr M. Acquainting this gentleman with the Nature of the business in hand he, like a good sportsman, fell into the scheme and at once volunteered to drive us to &quot;the best spot in the world.&quot; &quot;Bessie&quot; was harnessed to the village cart, and away we went at the rate of fully fourteen miles an hour in the direction of the Forest. I have sat behind many a good goer, but must for her size award the palm to Bessie a beauty barely twelve hands high. Throwing the reins in the bottom of the vehicle, Mr M. let the pony have her head, and the way she went uphill, down dale, and past other traps was truly astonishing. Great St. Peter's was passed, and lovely scenery encountered, the gardens being full of flowers, geraniums and roses blooming on all sides. An enjoyable drive of forty-five minutes found us at Boequaire [?] Bay, and directly in front stood the island of Lihou. As many of our readers may know, Lihou is a small hill of land that at low water can be reached on foot from Guernsey. As a fact, a paved roadway is laid in the bed of the ocean, across which both foot passengers and carts easily make their way. With the tide up communication is effected by boats, but this being low at the time we had the novel experience of   

Driving Through the Sea.   

Gaining the other side we &quot;explored&quot; and as the piece of land is only some three hundred yards by a hundred and fifty, we were not long about it. Two excellent patches of level turf were found, but just as we were making for the trap a gaunt, Robinson Crusoe-looking individual scaled one of the small stone walls and barred the way. Of course, he wanted to know our business, and, after a rather rough opening, quickly &quot;listened to reason.&quot; In the course of conversation we found that the population of the island was limited to three, the old &quot;josser&quot; and his two daughters, with a &quot;moke&quot; thrown in. Pointing a long skinny finger, which might have belonged to the Witch of Endor, towards the quadruped mentioned, the old man said, &quot;Us two keeps the family. He draws the seaweed and I burn it.&quot;  &quot;And why do you burn it?&quot; &quot;Well, to make [Col. 4c–5a] ashes, and from the ashes iodine is manufactured.&quot; &quot;And how much do you earn a week by that?&quot; said  I. &quot;Oh, about 4''s.''; not 5''s.'' any week.&quot; Great Scott! and this was all the old man, his two daughters,  and the &quot;moke&quot; had to live on. Handing him a franc each, which nearly stupefied the old chappie, Fleming and myself returned to Mr M., who drove us rapidly back to St. Peter's Port. The next day we crossed to Sark and Jersey, but found nothing so suitable as the ground at Lihou, and fairly well satisfied with the result of our visit, a day later boarded the Lynx for the return journey.   

During our visit the weather had been of a fairly favourable description, but the voyage home was made under very depressing circumstances. Rain fell heavily and a thick mist hung over the water, which after going a few miles was alive with porpoise. Some of these animals came within a few yards of the Lynx, but the speed of the boat proved too good for the finney ones, who were eventually left behind. With a breeze springing up sail was hoisted, and dashing through a heavy sea in splendid form the Lynx made some sixteen knots in the hour. In this way half the journey was completed, when all of a sudden we were caught in a squall. In an instant the wind chopped round, and before you could say &quot;Jack Robinson&quot; the good vessel was turned broadside to. The order to take in sail was promptly obeyed, but the clatter on deck brought many an anxious passenger rapidly from below. A very heavy sea was now running, and though the Lynx kept her course bravely the decks were swept from stem to stern. A huge East India-man now hove in sight, and, as one of the men informed me, she had the audacity to cross our bows instead of passing under our stern. This breach of ocean etiquette ruffled the skipper of the Lynx, who chased the trader, and eventually made her alter her course. This little &quot;diversion&quot; caused much amusement amongst our passengers, and after a cold, rough passage we landed safely at Weymouth. Chilled to the bone, we made for the Victoria Hotel, where a chop and a glass of old October revived the almost stagnant circulation. London was reached about 9.30, and, tired and worn out, I was glad to lay my head on a pillow I had not seen for four days.   

From London to Ostend.

Unforseen [sic] difficulties having cropped up, the Channel Island scheme had to be abandoned; consequently we were once more called upon to leave our native land. This time Belgium was our destination, and having carefully studied the map of that country, we took our departure one fine starlight evening from Charing-cross by the 8.15 Continental express. In the previous week we had the luxury of a compartment to ourselves, but on the occasion under notice we were not so fortunate. Two individuals, one with a strong Midland accent, and the other a Frenchman speaking bad English, shared the carriage with us, and at such a rate did the former's tongue go directly he entered that I gave up reading as a bad job, with a sigh put away my unfinished book of the previous trip. Fleming fared better, and having a knack of ignoring all around him, once more tackled &quot;Blood Money,&quot; and almost read himself off to sleep.  In the meantime our friends in the opposite corner made &quot;hay,&quot; and I soon found out they were both in the lace trade. With a tongue that went a couple of hundred to the minute, the Midlander almost monopolised the conversation, from which I gleaned the two were bound for Calais, and thanked goodness we should part at Dover.

Reaching this port about ten we boarded the Ostend boat, which, as regards passengers, was very poorly patronised. Hardly a breath of wind was stirring, and with the sea calm as a mill-pond we had a pleasant passage over, and landed safely at Ostend soon after 3 a.m. A cup of coffee was hastily swallowed, and a half-hour's run in the Brussels express found us  [4, Col. 5b]

At Bruges.

Leaving the train, we had considerable difficulty in finding our way out of the station. For a moment we caught sight of a drowsy old porter, but the last carriage had not passed away from the platform ere he was lost to view. Seeing an opening on other side we made for it, only to have our passage barred by a half-dressed Belgian who, whilst displaying a total ignorance of English, Irish, and Scotch, directed us to return the way we came. For some time we tried in vain to find an outlet, and just as I had made up my mind to walk the platform until daylight Fleming found an iron gate ajar. Without loss of time we were in the street, and crossing the round, knocked loudly at the door of the nearest hotel. All was in darkness, and with no response to our fourth appeal for admittance, a return to the station or a &quot;tramp round&quot; appeared to be our only alternative. A slight noise inside followed by a shuffling of feet, again raised hope within us, and when, after some trouble, the door swung back and brough to view a shock-headed Belgian our relief was great. No time, I can assure you, was lost in getting between the sheets, and rising betimes we again went on a voyage of discovery. Fleming had a chum in the town, and to his house of call our steps were directed. Again disappointment was in store for our wanderers, and a feeling of disappointment came over me when we found Mr G. was away. There was, under the circumstances, nothing left but to wait, and Fleming, who puts on hair rapidly, decided to place himself under the care of the town barber, or, I should, properly speaking, my ''coiffeur''. For this piece of indiscretion he had to come away minus a slice of his lower lip, which, fortunately, was not large enough to seriously reduce his avordupois [sic?], and heaping blessings (?) [sic] on mankind generally, not in any way forgetting the &quot;Bruges Barber,&quot; we roamed over the old town, noted for its magnificent cathedral and many seats of learning. The hours sped on pleasantly enough, and in due time our friend was unearthed. Being pledged to secrecy I am not going from my word by giving full particulars of

The Battle Ground

placed at our disposal. This is in close proximity to the town — in fact, not half an hour's walk, and on reaching the spot we were shown a lovely piece of turn. A clump of trees on each side partly hid the house from view, and when I say that the ring was to be pitched on a lawn tennis ground, it will be seen that neither side could claim an advantage as regards standing ground, &amp;c., the lawn being as level as a billiard table. Though our efforts so far had been crowned with success, we did not get permission to use the &quot;pitch&quot; without much trouble. The matter had to be seriously considered, and Mr G. would not give a decided answer without waiting another day. Consequently our stay had to be extended and the following morning, after repeated promises that the affair should be conducted in an orderly manner with not a rough at the ring-side, we got the answer, &quot;Yes.&quot; &quot;But,&quot; said Mr G., &quot;if there is any row or crowd I can assure you you will have a journey for nothing; therefore if trouble is at all probable keep away and save your money and time.&quot; As long as faith was not broken by either side we saw no reason why the matter should not be settled in a proper manner, and making all the necessary arrangements and laying down plans of a most complete description, we returned to town well satisfied with the result of a visit to Bruges. [4, Col. 5c]

Trouble about the Referee. 

On returning to town we received satisfactory reports of the work done by each of the contestants. Slavin, looked after by Josh Cosnett, the Birmingham heavy-weight, and his friend Mr Michael Taylor, of Australia, though rather light, was in good health. At one time he scaled only 12st 8lb, but a judicious spell of light work brought him back to 13st odd. Considering the importance of the business in hand, and its bearing on future events, Slavin had some cause for anxiety. In addition to this he was fighting for 300''l.'' of his own money, and consequently wished to do himself justice in the international battle. 

The next question was the selection of someone capable and willing to fill the all-important post of referee. According to the articles it was left to the stakeholder to appoint the official named providing the representatives failed to agree, and in the end this course had to be adopted, solely owing to the objections lodged by Smith. The first gentleman suggested by Slavin's side was Mr B. J. Angle, whom Smith promptly objected to. Next a well-known owner of racehorses (a gentleman who adjudicated upon the knuckle fight between George Camp and &quot;Click&quot; Soles) was proposed with no better result. Mr Bat Murphy, strange to say, would not suit, and then in succession &quot;No&quot; was said to the names of  Messrs G. H. Vise, A. F. Bettinson, and [B.?] Bayley. All these gentlemen, it must be stated, were proposed by Slavin and objected to by Smith, who in the meantime had named a member of the Pelican Club, a gentleman of limited experience in these matters, and the landlord of the house from which he was training, Mr W. Riley, Newmarket. This brought about a deadlock, and the state of affairs getting wind, ugly rumours were soon afloat, and even money, and in many cases good odds, were betted that there would be &quot;no fight.&quot; All this while Slavin was chafing and fuming at the delay, whilst at the same time his  representative and Smith's were at work almost night and day endeavouring to bring about a settlement of the difficulty. For a long time it looked as though the matter would end in smoke, as after the services of the stakeholder had been called upon great difficulty was experienced in getting anyone to accept the office of referee. So late as Friday at noon nothing had been done, and it was not until the evening of that day that Mr Joe Vesey, consenting to stand, was nominated by the stakeholder. 

A Scare.

Though the battle-ground was only known to two persons, the country where the fight was to be decided being, in fact, or supposed to be, a perfect secret, on Friday morning a paragraph said to have been sent out by a news agency appeared in several papers, and created intense astonishment and consternation. This, set out in large type by those who we should have thought would have suppressed the account, stated that Smith on the previous evening had arrived in London from his training quarters, and even went so far as to name the street he slept in. That Smith's visit to town was extremely injudicious there is not the slightest doubt, but this was not the worst. The time of departure for the scene of action was given, with a description of the place selected. This publicity had the effect of keeping away many who bad expressed a wish to be present, and on Slavin's side no less than eight gentlemen, each holding a high position in the sporting world, declined to have anything further to do with the affair. Who divulged the secret is at present a mystery, but there is not the slightest doubt that the information so freely put forward not only opened the eyes of the authorities, but gave the &quot;mob,&quot; to a certain extent, a key to the [Col. 5c–6a] situation. Slavin received the news with no small amount of astonishment, and a couple of telegrams informing him that a batch of the &quot;boys&quot; had gone across to make things warm for him did not give peace to his troubled mind. Smith made his way out of England on Friday by the short sea route, Dover to Calais, which he reached in safety. Slavin, not so fortunate, was arrested on Satuday, as already reported, but, after being bound over to keep the peace, he got away the same evening.&lt;/blockquote&gt;

==== Performance of the Men. ====
&lt;blockquote&gt;Jem Smith

was born in Whitecross-street, in the parish of Cripple-gate, on January 21, 1863. He stands 5ft 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in high, and weighs in condition 12st 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;?&lt;/sub&gt;in; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; hips, 41in; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in; calf, 16&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing, after one or two rough-and-tumbles with his mates he was taken in hand by Jack Knifton (&quot;81-tonner&quot;), and later by Mr F. Grimm, of the Central Club. Smith has never been defeated, though in his first engagement he did not shape like a coming Champion. His first success was in a competition at the Griffin, Shoreditch. Later he beat Snowey, of Holloway, and was then matched to box Harry Arnold, with gloves, for a 10''l.'' purse. Arnold, though a stone and a half lighter than Smith, stripped in good condition, having for some time previously been engaged in training Jack Massey for an important engagement. The affair was looked upon as a good thing for Smith, but it took him nearly an hour to polish off the Seven Dials' representative, though he had slightly the best of the encounter from start to finish. Subsequently the veteran Goode deposited a [sov?] on behalf of Smith to match him to fight Massey, but this was forfeited. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for 25''l.'' a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end twelve rounds. Smith also won Tom Symonds's All-comers' Competition on February 26[?], 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. At the close Smith was presented with a large oil painting of himself, executed by a prominent patron of the sport. Smith's next battle was with Jack Davis for 100''l.'' a-side and the Championship, on the turf, and came off on the borders of Surrey and Sussex, on Wednesday, December 17. In all six rounds were fought, occupying 7min 56sec, Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for 300''l.'' at Maisonne-Lafitte, near Paris, on Tuesday, February 16, 1886 [?]. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Jack Knifton, the &quot;81-tonner,&quot; as he is called. This proved a ''fiasco'', a two days' dreary tramp on the Continent being followed by a return to the old country. Knifton's people claimed that Smith had more than the specified number of followers, but there is little doubt that only one side wanted a fight, and that was not the &quot;81-tonner's&quot; lot. Meeting after meeting was held, Smith offering to fight anywhere the other liked, and after much wrangling arrangements were made to bring the affair off near Oxted [?]. Somehow of other the police got wind of it, and overtaking the Knifton party near Croydon rendered another postponement necessary. this time the arrangements were left in the hands of Knifton's side, with the result that the principals, seconds, and a dozen others were captured early one morning in a furniture van by the police before they had gone fifty yards as their journey. All were brought up at Thames Police-court, and after a remand bound over to keep the peace for twelve months. The memorial struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19, 1887 [?]. Jem's recent performance was a Ten-round Glove Contest with Jack Wannop, the Champion laying 250''l.'' to 200''l.,'' and winning easily, following which he was beaten in two rounds at the Pelican Club by Peter Jackson, the Coloured Champion, in a glove contest, for 1000''l.''  [4, Col. 6b] 

Frank P. Slavin. 

the Champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft 1in in his stockings, and in condition scales 13st 5lb or 13st 6lb. Slavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. He always had a liking for the gloves, and sparred a great deal when young, but did not enter into any match until the latter part of 1885 — that is, for money, though he won several competitions previously as an amateur. Slavin's first fight was with knuckles against M. Power for 50''l.'' a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with 2oz gloves for 200''l.'' a-side and the Championship of Queensland, winning in three rounds. This was in December, 1885 [?], and in the following year the Australian formed a combination and went touring, offering purses ranging from 10''l.'' to 20''l.'' to any man who stood up for four rounds. At this game Slavin had an easy time of it; some stood one round, and some two, but none of them four. Towards the end of the year he went to Sydney, and entered into an agreement with Larry Foley, who has a big place there. Here Slavin met Taylor, a pupil of Jackson, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big fellow named Bligh for 10''l.'' a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for 100''l.'' a-side. Dooley had a big reputation, but Slavin won in nine rounds. The subject of these remarks then had a little difficulty in finding an opponent, but eventually Mike Costello, an American, a good man, made his appearance, and the pair were matched for 100''l.'' a-side. They had a big battle in Sydney, lasting two hours and a half, and a draw was the result. Another match was made for 400''l.'', and 50''l.'' a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for 24''l.'' a-side to stop Costello in six rounds, but the latter is a very clever and tricky man, and lasted the time. Subsequently Frank journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for 100''l.'' a=side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for 250''l'' a-side and the Championship. Jackson's party wanted a match with gloves for a purse. They could not come to terms, though Slavin's backers offered Jackson 50''l.'' to sign articles for 500''l.'' Australian was now a bit bare, so in January last year the Australian made a trip to New Zealand. The first man put up was Snath, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the Champion. Terms could not be agreed upon until they had &quot;fiddled&quot; Slavin for nearly six weeks. The pair were then matched for 200''l.'' a-side, and fought with 2oz gloves at [Wangazul?], Laing failing to come up at the end of the fifth round. After this Slavin went back to Melbourne, hoping to get on a match with Jackson, and was surprised to find he had gone to America. Following this he met Jack Burke, but after Slavin had had all the best of it the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for 200''l.'' with small gloves, which ended in Mike being knocked out in 2min 40sec. He was very bad, and frightened all by not coming to for nearly half an hour. Slavin returned to Melbourne, and Burke having put several challenges in the paper, our visitor offered to stop him in six rounds; 200''l.'' a-side was put up and articles signed for 4oz gloves, but on the night Burke's party threw down regular pillow-cases. Of course this did not suit the other side, and there was a fine old wrangle, lasting two hours and a half. Eventually Mr Joe Thompson, who was referee, declared all bets off, and the two fought 6oz gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for 200''l.'', with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. This is considered to be one of Slavin's best performances. About this time Woolf Bendoff, of Whitechapel, was in Melbourne, and finding plenty of backers, they put up a forfeit of 100''l.'' for the Londoner to box Slavin, for 500''l.'' a-side. After seeing the latter polish off Burke, however, Bendoff's friends would go no further with the atch, and Slavin accordingly pocketed the &quot;century.&quot; An easy victory over Bill Buck in two rounds followed, and then Slavin, on October 17 made his first appearance in England at Astley's Theatre, where he beat &quot;Chesterfield&quot; Goode in five rounds in a match of twelve rounds, for 400''l.''  [4, Col. 6c] &lt;/blockquote&gt;

==== THE PELICAN CLUB AND THE FIGHT. ====
&lt;blockquote&gt;To the Editor of the Sportsman. 

Sir, — As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it. — Yours truly,  

Cecil Raleigh, Hon. Sec. 

Pelican Club, 21, Denman-street, W., Dec. 23. 

__________ 

We hear on good authority that it is the intention of the members of the Pelican Club to present Slavin, the Australian Champion, with a belt, in recognition of the manly exhibition he gave yesterday at Bruges. &lt;/blockquote&gt;

==== THE CHAMPIONSHIP OF THE WORLD. ====
&lt;blockquote&gt;Frank Slavin to John L. Sullivan. 

Frank Slavin, the Champion of Australia, is prepared to meet John L. Sullivan, under London Prize Ring Rules, for 500''l.'' or 1000''l.'' a-side and the Championship of the World. Slavin has lodged with us a sum of 100''l.'', and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr R. K. Fox, of New York, and the money lodged with him or the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; will lead to a match.&lt;ref name=&quot;:1&quot; /&gt; &lt;/blockquote&gt;

=== Dundee ''Evening Telegraph'' ===
This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:&lt;blockquote&gt;There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.&lt;ref&gt;&quot;London Gives the Colonist a Great Reception.&quot; ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&amp;page=3&amp;article=052&amp;stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.&lt;blockquote&gt;The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; 

Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot;Police!&quot; Smith was hurried from the ring; but Slavin and his friends refused to budge, and, after &quot;Time&quot; had been called, they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to Margate. Smith and his friends were left behind at Bruges.

Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec.

Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other.

Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.)

Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him.

Round 5. — A light round, Smith going down before a light blow. Time, 11sec.

Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead.

Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners.

Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground.

Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min.

Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together.

Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander.

Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec.

Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin.

Round 14 and last. — Just a second before &quot;Time&quot; was called, Slavin, remarking on the treatment he had received, said, &quot;Are you Englishmen, boys?&quot; This was received with derisive shouts from Smith's party, and then, in an instant, a cry of &quot;Police&quot; was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, &quot;Why don't you give me fair play?&quot; Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, &quot;I shall refuse to act any longer. This man&quot; — pointing  to Slavin — &quot;can't have fair play. I declare it a draw.&quot; Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there.

Here the fight came to an end, the referee exclaiming, &quot;I shall refuse to act any longer. I shall declare it a draw.&quot; There was considerable surprise among Slavin and his supporters, and the Australian said, &quot;I came 16,000 miles to win, and why don't you give a man a chance?&quot; After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine.

Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— &quot;As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it.&quot; 

Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. 

That Slavin has now a right to the title (empty though it be) of &quot;champion&quot; is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— &quot;Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight.&quot; It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. 

Arrival of Slavin at Margate. 

Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, &quot;See the conquering hero comes.&quot; Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, &quot;Well, Slavin, how have you fared?&quot; &quot;Very badly,&quot; responded the Australian. &quot;As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders.&quot; — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. 

The Pelican Club and the Fight. 

A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— &quot;In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business.&quot; 

It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. 

The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— 

&quot;In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, &quot;F. P. SLAVIN. 

&quot;Nayland Rock Hotel, Margate.&quot; 

Mr. Fleming's Statement. 

Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. 

The Preliminary Arrangements. 

The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the &quot;lost art&quot; was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the &quot;old style.&quot; Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. 

There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. 

Performances of the Men. 

Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. 

Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; waist, 36&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; hips, 41in.; thigh, 24&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in.; calf, 16&lt;sup&gt;3&lt;/sup&gt;/&lt;sub&gt;4&lt;/sub&gt;in.; biceps, 15&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton (&quot;eighty-one tonner&quot;) and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the &quot;eighty-one-tonner&quot; as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.&lt;ref&gt;&quot;The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring.&quot; ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&amp;page=1&amp;article=006&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== The ''People'' ===

On Sunday, 29 December 1899, the ''People''&lt;blockquote&gt;Disgraceful Scenes. Expressions of indignation.

The prize fight between Jem Smith, who poses as the fistic &quot;champion of England,&quot; and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a &quot;strong corner.&quot; The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that &quot;Smith did not lose&quot; proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known &quot;sports.&quot; Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper.

The Fight.

The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of &quot; Police!&quot; Smith was hurried from the ring, but Slavin and his friends refused to budge, and after &quot;Time&quot; had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of &quot;Foul,&quot; the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, &quot;I do not want to stay here to see you killed.&quot; Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate.

Slavin's Opinion of his Treatment.

The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there.

Indignation in London.

Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee.

The Pelican Club Meeting.

At the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of &quot;the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.[&quot;] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed.

Slavin Challenges J. L. Sullivan.

Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&amp;page=16&amp;article=150&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Sportsman'' ===
On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the &quot;testimonial&quot; fund for Slavin as well as some letters to the editor about the Slavin-Smith match:&lt;blockquote&gt;
A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). 

The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris.

The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.'']

* Lord Mandeville  2 2 0
* Lord Royston  2 2 0
* Lord James Douglas  2 2 0
* Hon. Michael Sandys  2 2 0
* Fred Kotchie  2 2 0
* Mr Ike White  2 2 0
* Mr W. Riley  2 2 0
* Mr C. Parsons  2 2 0
* Mr J. Kilby  2 2 0
* Mr Frank Hinde  2 2 0
* Proprietors of The Sportsman  2 2 0
* Mr. J. Purdey  2 2 0
* Mr. E. W. Bayly  2 2 0
* Messrs H. A. Domby and Son  2 2 0
* &quot;Caption Coe&quot;  2 2 0
* Mr C. W. Bulpett  2 2 0
* Mr P. W. Carr  2 2 0
* Mr W. Morley Pegge  2 2 0
* W. D. S.  2 2 0
* J. E. S.  2 2 0
* Mr W. H. Grenfel  2 2 0
* Mr Quartermaine East  2 2 0
* Mr H. H. Hungerford  2 2 0
* Mr Noel Fenwick  2 2 0
* Sir John Leslie, Bart.  2 2 0
* Mr T. Brinkman  2 0 0
* Mr Calder  1 1 0
* Mr H. W. Hill  1 1 0
* Mr John Young  1 1 0
* Mr D. A. Neilson  1 1 0
* Mr W. Neilson  1 1 0
* Mr Julius Simon  1 1 0
* Mr J. Edge  1 1 0
* Mr H. Marshall  1 1 0
* Mr F. Aynsley  1 1 0
* Col. Hon. Wellesley  1 1 0
* Mr Charles Hawtrey  1 1 0
* Mr W. F. Ashton  1 1 0
* Mr Barratt  1 1 0
* Old Durham Ox  1 0 0
* Sir John Willoughby  1 0 0
* A friend, per C. H. A.  0 10 6
* Mr Ben Giles  0 10 6
* Mr Tom Vale  0 10 6
* [E.?] W.  0 10 6
* Officer  0 10 0
* Per Mr Ike White, Exchange Club:
** S. W.  2 2 0
** Mr G. B. Knight  1 1 0
** Mr A. Powles  1 1 0
** Mr S. Blandford  1 1 0
** T. F. W. C.  1 1 0
** Mr Leo Harward  1 1 0
** Mr J. Harward  1 1 0
** Mr H. Harward  1 1 0
** Mr [Monteagle?] [Moutcagie??]  1 1 0
** Mr C. Orate  0 10 6
** W. G. P.  0 10 6
** H. R. T  0 10 6
** Mr Josiah Smith  0 10 6
** W. H. R.  0 10 6
** Mr R. Wilson  0 5 0
* Per Mr M. J. Harlow, Margate:
** Mr A. H. [Cosi? Coni?]  2 2 0
** Mr J. Thurston  1 1 0
** Mr J. A. Fagg  1 1 0
** Mr F. [Miles?]  1 1 0
** Mr Adolphe [Drineqbeir?]  1 1 0
** Mr Ernest [Drineqbier?]  1 1 0
** Mr Lacy Brown  1 1 0
** Mr Algy Brown  1 1 0
** Mr M. J. Harlow  1 1 0
** An Admirer  1 1 0
** Mr J. W. Robinson  1 1 0
** Mr W. Searles  1 1 0
** Mr J. Smith  1 1 0
** Mr J. Hume  1 1 0
** Mr J. Painter  1 1 0
** A Scotchman  1 1 0
** Mr J. Mundell  0 10 6
** Mr Bob Redman  0 10 6
** Mr G. A. Body  0 10 6
** Mr J. Phipps  0 10 6
** Mr J. Doughty  0 10 6
** Old Fisherman  0 10 6
** Mr Joe Knott  0 10 6
* Per Mr B. J. Angle
** Some mebers of the Stock Exchange  125 0 0
** Hr Henry J. Turner  2 2 0
* Per Mr W. J. Innes:
** Mr Sidney Fry  2 2 0
** Mr W. J. I.  2 2 0
** Mr W. Schlesinger  2 2 0
** Mr Joseph Thompson  2 2 0
** Mr Barney Thompson  2 2 0
** Mr L. Meyers  2 2 0
** Mr W. Shoe  2 2 0
** Mr George Cooper  2 2 0
** Mr J. T. Crossley  2 2 0
** Mr Andy Anderson  2 2 0
** Mr G. M. T.  2 2 0
** Mr F. R.  2 2 0
** Mr W. Bertram  1 1 0
** Mr J. W. Hammon  1 1 0
** Mr Alec Harris  1 1 0
** Mr John Isaacs  1 1 0
** Mr Edward [Dance? Dasce?]  1 1 0
** Mr George Blanchett  1 1 0
** Mr F. [Leleu? Lolou?]  1 1 0
** Mr W. J. R.  1 1 0
** Mr H. C.  1 1 0
** Mr A. A. M.  1 1 0
** Mr D. W. Fowler  0 10 6
* Per Mr F. H. Gale:
** Mr R. Moore  2 2 0
** Mr A. H. Goode  2 2 0
** Mr [B.? E.? R.?] J. G. Goode  2 2 0
** Mr F. H. Gale  2 2 0
** Mr H. Ulph  1 1 0
** Mr J. Hooper  1 1 0
** Mr C. Kittall  1 1 0
** Mr C. Moore  1 1 0
* Per Mr W. Riley:
** Mr Brand (Wiltshire)  2 2 0
** Mr Charles Godfrey  2 2 0
** Mrs Lane (Britannia)  2 2 0
** Mr Sam Adams  2 2 0
** B. and M.  0 7 6
* Per Mr A. Cockburn:
** Mr Richardson  2 2 0
** Mr A. Cox  1 1 0
** Mr Hy. Brackenbury  1 1 0

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and &quot;knuckle-dustered&quot; him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, &quot;Come and have a round.&quot; The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called &quot;seconds&quot; should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of &quot;Police&quot; was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska &quot;tough,&quot; as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how &quot;seconds&quot; — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a &quot;duster&quot; might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &amp;c.,

A Lover of Boxing.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &amp;c.,

Leicester, December 28. J. Barrett.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;.

Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded.

I cannot, however, agree with those who hope &quot;that we have seen the last of prize-fighting,&quot; and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &amp;c.,

J. W. B. C., Junior Carlton Club, December 27.

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself &quot;England's Champion.&quot; I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon &quot;British fair-play and courage.&quot; — Yours, a lover of it, Sydney Galvayne. 

York, December 26. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, 

Birmingham, December 27. Tom B. Sharp. 

To the Editor of T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt;. 

Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &amp;c., 

I. White.  

Exchange Club, Queen-street, Cheapside, Dec. 28.&lt;ref&gt;&quot;The Ring. The Slavin Testimonial.&quot; ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&amp;page=4&amp;article=020&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt; &lt;/blockquote&gt;
On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:&lt;blockquote&gt;JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.
{{paragraph break}}

J&lt;small&gt;OHN&lt;/small&gt; L S&lt;small&gt;ULLIVAN&lt;/small&gt; has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me.&quot;&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:&lt;blockquote&gt;THE SLAVIN TESTIMONIAL
{{paragraph break}}
The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T&lt;small&gt;HE&lt;/small&gt; S&lt;small&gt;PORTSMAN&lt;/small&gt; Office.&lt;/blockquote&gt;
{| class=&quot;wikitable&quot;
!
!£
!s.
!d.
|-
|Amount already acknowledged
|407
|3
|6
|-
|Mr E.Tabernacle
|1
|1
|1
|-
|H. S.
|1
|1
|0
|}
&lt;blockquote&gt;The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.&lt;ref&gt;&quot;The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns.&quot; ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&amp;page=4&amp;article=049&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Huddersfield Daily Chronicle'' ===
On 10 January 1890:&lt;blockquote&gt;
Mitchell and Lord Mandeville.

As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of &quot;sports&quot; at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by &quot;Pony&quot; Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &amp;c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. &quot; [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and &quot;went for&quot; Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and &quot;Pony&quot; Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was &quot;something up,&quot; and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.&lt;ref&gt;&quot;The Smith-Slavin Fight.&quot; ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&amp;page=4&amp;article=071&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;
After the Pelican Club meetings, Slavin comments:&lt;blockquote&gt;On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English &quot;champion,&quot; made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.&lt;ref&gt;&quot;Slavin at the Royal Acquarium.&quot; ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&amp;page=3&amp;article=042&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== ''Reynolds's Newspaper'' ===
''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:&lt;blockquote&gt;Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. &quot;I can't see how an English fighter can be beaten,&quot; he said. &quot;This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of &quot;heelers&quot; always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me.&quot;&lt;ref&gt;&quot;The Recent Prize Fight.&quot; ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&amp;page=6&amp;article=053&amp;stringtohighlight=mandeville+viscount. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Cork Constitution'' ===
About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:&lt;blockquote&gt;This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s &quot;lefts.&quot;&lt;ref&gt;&quot;Mr. Abington.&quot; ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&amp;page=7&amp;article=099&amp;stringtohighlight=mandeville+viscount. Same title and p.&lt;/ref&gt;&lt;/blockquote&gt;

=== The ''Colonial Standard and Jamaica Despatch'' ===
This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:&lt;blockquote&gt;The disgraceful affair, called by courtesy and [sic] &quot;International Prize Fight,&quot; which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The &quot;fight&quot; lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn [??, s/b Royston?], Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.&lt;ref&gt;&quot;London.&quot; ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&amp;page=2&amp;article=019&amp;stringtohighlight=mandeville+viscount. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;
== Notes and Questions ==
# 

==References==
{{reflist}}</text>
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    <title>Growth per Nobel economists and others</title>
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      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
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      <comment>add Botswana + ...</comment>
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      <text bytes="64772" sha1="jdep3vjdff4fyuzd39pyo0vlrtlxn3b" xml:space="preserve">{{Research project}} 
:''This article invites all to contribute references and interpretations to the growth in average annual income, Gross Domestic Product (GDP) per capita at purchasing power parity, as applied to leading countries in the [[w:Maddison Project|Maddison Historical Statistics Project]].''
:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”'''{{'}} page]] -- and treating others with respect.''  
::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]''

== Abstract ==

This article first identifies and plots GDP per capita for leading countries in the Maddison Project data then discusses comments by Nobel prize winning economists and others to interpret what appears in said plot. No one really knows how to make economies grow. However, growth at the technology frontier requires creative destruction, and immigrants tend to be more entrepreneurial than native born. Also, creative destruction is facilitated and inequality is limited by diversity of ownership and control of the major media. 

== Maddison Historical Statistics == 

[[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD = initially [[w:Holland|Holland]] to 1807 and the [[w:Netherlands|Netherlands]] after; GBR = initially England to 1700, Great Britain to 1870, then the UK; USA starting in 1650 with the 13 colonies that became the US; SGP = Singapore starting in 1870).&lt;ref&gt;See Bolt and van Zanden (2024) for the Maddison Data generally, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, Sugimoto (2011) for Singapore to 2007, van Zanden and van Leeuwen (2012) for the data on Holland 1349–1807, and Smits et al. (2000) for the data on the Netherlands 1808-1913.&lt;/ref&gt; The dramatic drop in the orange line between 1807 and 1808 is due to the change in reference from Holland to the Netherlands; at that time Holland was the most prosperous region of the Netherlands.&lt;ref name=vanZ_B&gt;Private email exchange with van Zanden and Bolt, 2026-01-08.&lt;/ref&gt;'']]

The [[w:Maddison Project|Maddison Historical Statistics Project]] (aka Maddison Project) is widely recognized as the premier data source on historical macroeconomic data. It is named after [[w:Angus Maddison|Angus Maddison]] (1926-2010), who was a pioneer in [[w:National accounts|national accounts]], especially estimating [[w:Gros Domestic Product|gross domestic product]] (GDP) per capita (GDPpc) and population (pop) at various times dating back to the year 1 of the [[w:Common Era|Common Era]] (CE / AD). After Maddison's death, the Maddison Project has continued to update the database to include the best sources available on any country for any historical period since year 1. 

When accessed 2026-10-10 the version of the data available had been released 2024-09-18 containing data to 2022.&lt;ref&gt;Bolt and van Zanden (2024).&lt;/ref&gt; This dataset includes 23,280 observations on at least one of GDPpc and pop for an average of 137 years for 169 countries. Only 18 countries had the highest GDPpc for at least one year. Annual data starts in 1252, and that's only for one country, GBR = England / Great Britain / the United Kingdom. NLD (Holland / the Netherlands) had the highest GDPpc for 97 percent of the years between 1349 and 1807. The drop in the orange line in Figure 1 between 1807 and 1808 is due to the change of reference from Holland to the Netherlands.&lt;ref name=vanZ_B/&gt; Visual comparison of the orange and red lines in Figure 1 suggests that the GDPpc of Great Britain overtook that of the Netherlands a few decades earlier and overtook Holland a few decades later. This analysis focuses on data since 1349, in part because data were available for at most 4 countries for most years prior to that date. (The 3-letter country codes used here are [[w:ISO 3166-1 alpha-3|ISO 3166-1 alpha-3]], used in the Maddison data.) 

We are interested in technology leaders, so we eliminate the [[w:petrostates|petrostates]] of Qatar (QAT), Kuwait (KWT), and United Arab Emirates (ARE), plus Norway (NOR); the latter owes a substantial portion of their wealth to democratic management of [[w:North Sea oil|North Sea oil]].&lt;ref&gt;Dośpiał-Borysiak (2017).&lt;/ref&gt; With these restrictions 12 countries had the highest GDPpc for at least one year. However, FRA (France), ITA (Italy), SWE (Sweden), BEL (Belgium), NZL (New Zealand) and CHE (Switzerland) led for at most 7 of the 674 years between 1349 and 2022; we eliminate those, because including them would seem to add more clutter than clarity. We also eliminate AUS (Australia), which led for 17 of the 39 years between 1853 and 1891, when it was competing for the leadership with GBR and USA, for the same reason. And we eliminate LUX (Luxembourg), which has a population under a million. 

This leaves NLD, GBR, USA, and SGP (Singapore), plotted in Figure 1; Singapore has led since 2010.&lt;ref&gt;More details of this analysis are available in the chapter on &quot;[[Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs|The impact of the media on political economy since the time of the Pharaohs]]&quot;&quot; in the book-in-progress on ''[[Media Literacy and You]]'' on Wikiversity.&lt;/ref&gt; This plot has some interesting features, which we discuss in the remainder of this research report. 

== Nobel economists and others ==

[[w:Abhijit Banerjee|Banerjee]] and [[w:Esther Duflo|Duflo]] (2019, p. 207) said, &quot;despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive.&quot; That appeared shortly before they shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]] with [[w:Michael Kremer|Michael Kremer]] for documenting how [[w:Randomized controlled trial|randomized controlled trial]]s could be used to alleviating poverty in [[w:Development economics|development economics]].&lt;ref&gt;For the present discussion, we assume that enough [[w:Externality|externalities]] have been sufficiently internalized with appropriate [[w:Pigouvian tax|Pigouvian tax]]es to make growth desirable. This converts discussions of [[w:Degrowth|degrowth]], e.g., by Alexander (2026), into evaluations of whether externalities have actually been sufficiently internalized and what to do if they are not. Internalizing externalities appropriately will not be easy as long as those who benefit from not paying for damages they cause to customers, employees and neighbors control substantial portions of the funds for the media. However, for the present discussion, we assume this is not an issue.&lt;/ref&gt; 

While Banerjee and Duflo insisted that no one knows how to make an economy grow, other leading economists have recommended partial solutions. For example, [[w:Joel Mokyr|Mokyr]] (2017, p. 53) said, &quot;The willingness of the Netherlands and later Britain and the United States to tolerate Jews and dissenting Christians contributed a great deal to their economies, especially in high-skilled manufacturing and financial services.&quot; In other words, immigrants made an outsized contribution to the growth of NLD, GBR, and US, visible in Figure 1. Mokyr's comment appeared over a year before Banerjee and Duflo (2019) and roughy eight years before Mokyr shared the 2025 Nobel Memorial Prize in Economics with [[w:Philippe Aghion|Aghion]] and [[w:Peter Howitt (economist)|Howitt]]. 

Mokyr (2017, p. 255) also said that, &quot;intellectual innovation will eventually end up being increasingly resisted by an incumbent technology; if that resistance is not overcome, technological progress may be extinguished.&quot; That resistance, he suggested, drives {{quote|

the empirical regularity that no society remains at the cutting edge of technological creativity for very long&lt;ref&gt;Mokyr (2017, p. 260).&lt;/ref&gt;.}}

Mokyr (2017, p. 260; 1994) called this &quot;Cardwell's Law&quot; after Cardwell (1972), which describes {{quote|

the interaction between technological creativity and political competition ... based on the observation that technological progress encounters resistance from various groups that believe they stand to lose from innovation. These pressure groups will try to manipulate the political system to suppress successful innovation.}}

[[w:Robert Michels|Michels]] (1911) called this &quot;the [[w:Iron law of oligarchy|iron law of oligarchy]]&quot;. Further documentation of this appears in Acemoglu and Robinson (2012) and Acemoglu and Johnson (2023), who shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]]. 

Figure 1 begins with growth in England and Holland between 1349 and 1400, likely due to the fact that the [[w:Black Death|Black Death]] (1346-1353) killed between 30 and 60 percent of the population: Since land was not destroyed, survivors were likely able to produce more agricultural products for the same labor. Also, under the feudal system in place before the Black Death, commoners were not allowed to move without the permission of their employer, the Lord of the land where they worked. Efforts by elites to prevent commoners from moving and obtaining higher wages failed in England.&lt;ref&gt;Acemoglu and Robinson (2012, esp. pp. 99-101) note that surviving commoners in England were able to bargain for higher wages. The authorities passed emergency legislation in 1349 and 1351 that made it a crime to offer or accept wages higher than before the plague or to refuse to work or break an existing contract. In 1361 the penalties were increased to include imprisonment and even branding. This led to [[w:Peasants' Revolt|a major insurrection in 1381]] led by [[w:Wat Tyler|Wat Tyler]], which was put down with over 1,300 deaths. After that insurrection, authorities in England allowed commoners to demand and accept higher wages. Commoners were also allowed to move without the permission of their Lord, which had ''not'' been accepted before. By contrast, the treatment of commoners in Russia was reportedly ''worse'' than before the plague. Changes in the treatment of commoners between England and Russia seemed to vary depending on proximity to these extremes, with Austria more like Russia and France more like England.&lt;/ref&gt; The growth in Holland from 1492 has been has been attributed to [[w:Expulsion of Jews from Spain|Jews expelled from Spain settling]] in Holland and starting new businesses. 

The iron law of oligarchy provides an explanation for why growth stopped in England around 1400, Holland around 1600, and why growth has been slowing in recent decades in all the countries plotted in in Figure 1. It can also explain [[w:History of the Roman Empire|the rise and fall of the Roman Empire]], of the [[w:Maya civilization|Maya civilization]], and other societies that became very rich from allowing commoners to become entrepreneurs and then declined after previous winners changed the rules of the political economy to block further [[w:creative destruction|creative destruction]]. This includes how [[w:History of the Republic of Venice|Venice]] became &quot;possibly the richest place in the world, with the most advanced set of inclusive economic institutions&quot;, dominating trade in the [[w:Mediterranean Sea|Mediterranean]] based on a system that encouraged entrepreneurial commoners to undertake risky trading ventures starting after 810 CE and ending when leaders made wealthy by such ventures changed the rules to end that system starting in 1297, destroying their dominance of Mediterranean trade &quot;and converting Venice into a museum&quot;, in the words of Acemoglu and Robinson (2012, pp. 152-155).

Acemoglu and Robinson (2012) suggest three primary distinctions between the poorest and wealthiest nations. The poorest tend to be [[w:failed state|failed states]] like [[w:Somalia|Somalia]], which are poor, because of a substantial lack of rule of law and continued violent conflict between warring groups. Acemoglu and Robinson divide other nation states between &quot;extractive&quot; and &quot;inclusive&quot; regimes. &quot;Extractive&quot; regimes tend to be poor, because elites write the rules to protect their interests against the competition that generates the creative destruction required for economic growth at the technology frontier. &quot;Inclusive&quot; regimes limit monopolies and encourage enough commoners to become entrepreneurs that over time the economy tends to produce more with less, increasing the wealth of the nation. 

Trainer (2025) described the 2024 Nobel Prize for Economics as, &quot;An example of how conventional economics misrepresents reality&quot;, blaming the poor for their poverty, e.g., in failing to cite Perkins (2004) ''Confessions of an Economic Hit Man''. Doth professor Trainer protest too much? Chapter 9 in Acemoglu and Robinson (2012) describes, &quot;Reversing Development: How European colonialism impoverished large parts of the world&quot;, which describes several historical cases where a colonial power subordinated a local extractive elite, increasing the severity of their extractive policies, further impoverishing the vast majority of that locality. The Dutch East India Company did this to most of Indonesia, but they could not do this to the Banda islands, which consisted of many small autonomous city-states, run by village meetings of citizens with no central authority whom the Dutch could coerce into signing a monopoly treaty. To avoid having to compete with English, Portuguese, Indian, and Chinese merchants, the Dutch massacred &quot;almost the entire population of the islands, probably about fifteen thousand people. ... [O]nly a few were left alive, enough to preserve the know-how necessary for mace and nutmeg production. After this genocide was complete,&quot; the Dutch repopulated the Banda islands with a hierarchical plantation society.&lt;ref&gt;Acemoglu and Robinson (2012, p. 248).&lt;/ref&gt; 

[[w:Botswana|Botswana]] was one of the few places in the colonized world that successfully avoided this fate. In 1895 chiefs of three of the eight Tswana states in what is now Boswana sailed to England with a charm offensive to save their country from Cecil Rhodes. They met with Joseph Chamberlain, then colonial secretary, before and after an extensive speaking tour of England. With the favorable coverage that the chiefs got in the English press, Rhodes got a railroad through their country and nothing more. When Botswana gained independence in 1966, they had relatively inclusive political institutions that allowed them to grow economically, escaping the fate of Black Africans in Rhodesia / Zimbabwe to their north and South Africa to their south.&lt;ref&gt;Acemoglu and Robinsion (2012, pp. 404-405).&lt;/ref&gt; 

Robinson (2025) provided a related explanation for the poverty of [[w:Zimbabwe|Zimbabwe]], which has a 2026 population of roughly 17 million souls with a 2025 GDP per capital of roughly $5,400 at PPP. '[D]uring fieldwork in Zimbabwe in 2017 I had breakfast with the CEO of a subsidiary of a large multinational company. I was trying to understand the political incentives and mechanisms that had led to extractive economic institutions in the country, leading it to be substantially poorer today than it was at the time of independence in 1980. ... At some point in the conversation the CEO looked up at me and said: “I really don’t know what your problem is, it really isn’t complicated, what is it that you can’t understand? ... [T]here’s plenty of money in this country for 300 people.”'

Mokyr (2017, p. 60) also mentioned &quot;entrepreneurs of hate&quot;, who convince others to dislike some group in a way that may benefit those &quot;entrepreneurs&quot; and major media organizations, who promote such &quot;entrepreneurs&quot; to please those who control much of the money for the media. For-profit social media have been particularly effective in herding their audiences into echo chambers, reinforcing their preconceptions in ways that increase political polarization and violence and threaten democracy.&lt;ref&gt;See, e.g., DiResta (2026), DiResta and Keinfeld (2025), and the Wikiversity article on &quot;[[Managing the good and the evils of social media]]&quot;, accessed 2026-10-14.&lt;/ref&gt; When successful this protects current elites from competition that would otherwise drive [[w:creative destruction|creative destruction]], benefitting the public at large while threatening existing elites.  

For example, most research on immigrants suggest that on average immigrants are more entrepreneurial and no more likely to commit crime than native born,&lt;ref&gt;e.g., Martínez et al. (2017); Muchow and Stanishevska (2026); Ajzenman(2023).&lt;/ref&gt; and some studies report that [[w:Sanctuary city|sanctuary jurisdiction]]s have on average higher median incomes and less crime.&lt;ref&gt;Martínez et al. (2017); Wong (2017). See also the Wikipedia article on &quot;[[w:Sanctuary city|sanctuary city]]&quot;, accessed 2026-10-12.&lt;/ref&gt; Where immigrant crime is higher, it typically result from anti-immigrant labor market restrictions or from young less-educated males with irregular status.&lt;ref&gt;Marie and Paolo (2024). See also the Wikipedia article on &quot;[[w:Immigration and crime|Immigration and crime]]&quot;, accessed 2026-10-12.&lt;/ref&gt; 

But major media outlets make money from featuring charismatic figures, who attract an audience without offending people who control much of the money for the media. In 2017 President [[w:Immigration policy of the first Trump administration|Trump]]'s Attorney General [[w:Jeff Sessions|Jeff Sessions]] quoted a research study that supposedly showed that sanctuary cities had more crime than cities without sanctuary policies. The authors of that study insisted that Sessions had ''misrepresented'' their work, which concluded &quot;that sanctuary policies have no effect on crime rates, despite narratives to the contrary.&quot;&lt;ref&gt;Roll (2017); Gonzalez O’Brien, et al. (2017).&lt;/ref&gt; Similarly, a 2017-09-19 report described how, &quot;The Trump administration rejects its own study finding refugees help the economy&quot;. &lt;ref&gt;Swanson (2017).&lt;/ref&gt; Just under three months later, the [[w:Union of Concerned Scientists|Union of Concerned Scientists]] (UCS) described how, &quot;The Trump Administration Manipulates Study on Economic Benefits of Refugees in US&quot;: The [[w:Refugee Act of 1980|Refugee Act of 1980]]  requires the president to set by October 1 the minimum number of refugees to be allowed into the US the following year. To support this, the [[w:Department of Health and Human Services|Department of Health and Human Services]] (HHS) produced a [[w:Cost–benefit analysis|cost-benefit analysis]] on the economic impact of refugees. A draft of this 2017 analysis concluded that “the net financial impact of refugees was positive over the 10-year period, at $63 billion.” Then [[w:Stephen Miller|Stephen Miller]] intervened to delete the benefits -- tax revenues -- generated by refugees. The resulting final report concluded that the &quot;per-capita refugee costs for major HHS programs [averaged over 10 years] totaled $3,300.” Miller’s [[w:United States Domestic Policy Council|Domestic Policy Council]] (DPC) reportedly &quot;maneuvered around the [[w:United States Department of State|State Department]], [[w:United States Department of Defense|Defense Department]], [[w:Joint Chiefs of Staff|Joint Chiefs of Staff]], [[w:Office of the Vice President of the United States|Office of the Vice-]]President, and [[w:Office of Management and Budget|Office of Management and Budget]] to push through his draconian cuts to the refugee cap, dismissing and rewriting the evidence as needed.&quot; A State Department official said, &quot;The process has never been this corrupt.” UCS reported that, &quot;Manipulating scientific evidence to fit a political narrative is a clear path toward implementing policies that are detrimental both to the US and to our allies.&quot;&lt;ref&gt;Union of Concerned Scientists (2018).&lt;/ref&gt;

To the extent that these comments provide a balanced summary of the research in this area, it suggests that leaders like Trump, Sessions, and Miller, to name only three, can be fairly describes as &quot;entrepreneurs of hate&quot;. 

We next discuss competition in media markets before discussing immigrants further. 

=== Diversity of ownership of the major media === 

Acemoglu and Johnson (2023, esp. ch. 4) describe how religious leaders in hierarchical societies prior to [[w:James VI and I|King James]] of the [[w:King James Version|King James Bible]] convinced commoners to live in poverty while giving increasing shares of what they produced, so religious and secular elites could live in opulence. Prior to King James the major media were religious organizations. In such hierarchical societies some religious leaders were entrepreneurs of hate, exhorting the faithful to fear and hate designated evil others and give increasing shares of what they produce so secular elites would &quot;protect&quot; them in wars against designated evil others.

Acemoglu and Robinson (2012, p. 461) said, &quot;Empowerment of society at large is difficult to coordinate and maintain without widespread information about whether there are economic and political abuses by those in power. ... Pamphlets and books informing and galvanizing people played an important role during the [[w:Glorious Revolution|Glorious Revolution]] [1688] in England, the French Revolution [1789-1799]&quot; and in the US. ([[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]] and [[w:James A. Robinson|Robinson]] shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]].)

But pamphlets started challenging established power structures much earlier. The publication of [[w:Martin Luther|Martin Luther]]'s ''[[w:Ninety-five Theses|Ninety-five Theses]]'' in Latin 1517-10-31 was quickly condemned by [[w:Pope Leo X|Pope Leo X]] and his hierarchy. [[w:Johann Tetzel|Johann Tetzel]], who had been selling [[w:Indulgence|indulgence]]s, published a pamphlet calling for Luther to be burnt for [[w:heresy|heresy]]. Eight hundred copies were reportedly sent to [[w:Wittenberg|Wittenberg]] to be sold, but students seized and burned them.&lt;ref&gt;Hendrix (2015, p. 64), quoted from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; This led to a pamphlet war. Luther's brief German-language reply was short and easy for commoners to understand&lt;ref&gt;Leppin and Wengert (2015, p. 389), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; and was much more popular than Tetzel's.&lt;ref&gt;Pettegree (2015, p. 145), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; 

Less than twenty years later, in 1534, King [[w:Henry VIII|Henry VIII]] declared himself head of the [[w:Church of England|Church of England]], denying the authority of the Pope. To secure the support of his subjects, he drove a propaganda war that included print literature&lt;ref&gt;Raymond (2006), cited from the Wikipedia article on &quot;[[w:Pamphlet war|pamphlet war]]&quot;, accessed 2026-10-15.&lt;/ref&gt; and theatre and minstrel troupes funded by the king. The latter traveled around the land to promote the new religious practices. Henry was portrayed as the glorious king of England, bravely and heroically defending the true faith against foreign devils, especially the Pope and Catholic priests, monks, etc.&lt;ref&gt;Hibbert et al. (2010, p. 928), cited from the Wikipedia article on &quot;[[w:Henry VIII|Henry VIII]]&quot;, accessed 2026-10-15.&lt;/ref&gt;

Henry died in 1547. Less than a dozen years later, the crown passed to [[w:Elizabeth I|Elizabeth I]]. Her limited personal wealth and English tradition required her to negotiate with Parliament for tax money. Acemoglu and Robinson (2012, pp. 105-111) noted that she was forced to limit the granting of monopolies and allow a class of merchants to compete in international trade. Many English merchants became wealthy, helped by the defeat of the [[w:Spanish Armada|Spanish Armada]] in 1588. 

When Elizabeth died in 1603, she was replaced by her cousin, King [[w:James VI and I|James VI of Scotland]], who then became also James I of England. He claimed [[w:Divine right of kings|divine right of kings]], threatening this class of merchants. To raise money without asking Parliament, he granted patents and monopolies to those who would pay. Parliament responded by passing the [[w:Statute of Monopolies|Statute of Monopolies]] on 1624-05-29. This statute repealed some past and future patents and monopolies except for novel inventions. 

But that Statute of Monopolies did not end the conflict. It ultimately went public, described as, &quot;A Pamphlet War in England, 1641-1643&quot;.&lt;ref&gt;Washington (2018).&lt;/ref&gt; Some pamphlets supported [[w:Charles I of England|Charles I]], who became King after James died in 1625. Others supported Parliamentarians and [[w:Levellers|Levellers]]. The latter supported radical ideas like [[w:popular sovereignty|popular sovereignty]], extended suffrage, equality before the law and religious tolerance. In 1642 war broke out between Parliament and Charles I, though many in Parliament sided with Charles -- especially those, who had received monopolies from him.&lt;ref&gt;Acemoglu and Robinson (2012, esp. pp. 189, 210).&lt;/ref&gt; The war was won by the [[w:New Model Army|New Model Army]], which represented many in Parliament plus Levellers. The latter were excluded from the government that followed the [[w:Execution of Charles I of England|beheading of King Charles I]] in 1649.&lt;ref&gt;Foxley (2013, p. 207), cited from the Wikipedia article on [[w:Levellers|Levellers]], accessed 2026-10-15.&lt;/ref&gt; However, that new government stopped granting as many monopolies and allowed so many commoners to become entrepreneurs that the GDP per capita began to grow at rates that are key features of Figure 1 for England and the 13 English colonies that became the US over a century later. 

One explanation for that economic growth was the diversity of independent publishers of pamphlets, books and newspapers. [[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]] of 1792&lt;ref&gt;Wikipedia &quot;[[w:Postal Service Act|Postal Service Act]]&quot;, accessed 2023-07-11.&lt;/ref&gt; made a major contribution to making the US what it is today.  Under that act, newspapers were delivered up to 100 miles for a penny, when postage for letters was between 6 and 25 cents depending on distance.&lt;ref&gt;Historian, USPS (2010).&lt;/ref&gt; McChesney and Nichols estimated that between 1840 and 1844, the US postal subsidy was a little over 0.2% of GDP.&lt;ref name=McC-N2010&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; 

During the first half of the nineteenth century the US had more independent newspaper publishers (more independent media organizations) per million population than at any other time or place in human history.&lt;ref&gt;John (1995). See also the interview with him at &quot;[[Media concentration per Columbia History Professor Richard John]]&quot; on Wikiversity, accessed 2026-10-12.&lt;/ref&gt; In 1932 with official unemployment over 20 percent, the conservative argument that blames the poor for their poverty did not sell newspapers, and the US still had sufficient competition in media markets that presidential candidate Franklin Roosevelt (FDR) got a mandate to spend the money to put humans back to work while taxing the ultra-wealthy like they had never been taxed before or since and during World War II to institute wage and price controls under the [[w:Office of Price Administration|Office of Price Administration]]. The latter prevented price gouging by government contractors that had generated inflation and stifled economic growth in previous major wars (World War I, the Civil War, and the War of 1812). Instead, the US averaged 7 percent per year growth for FDR's 12 years.&lt;ref&gt;The &quot;6 percent per year&quot; growth during FDR's 12 years was computed as the average of log(GDPpc) in the Maddison Data. To be precise, that computation produced 0.0597; converting that from a log scale to percent increase gives 0.0615. Using data from [[w:MeasuringWorth|MeasuringWorth]] instead of Maddison, we get 0.783 and 0.815, respectively. More documentation of this is available in &quot;[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]&quot; on Wikiversity, accessed 2026-10-13.&lt;/ref&gt; 

One of the most interesting features of Figure 1 is the behavior of the US economy in the administrations of [[w:Herbert Hoover|Herbert Hoover]] and FDR: Under Hoover, the economy declined at 9 percent per year for 4 years, a financial disaster that created the public demand for change that gave FDR the mandate to do what he did. Economic growth under two other presidents are numerically comparable to FDR's numbers but for much shorter periods: (1) [[w: Warren G. Harding|Warren G. Harding]] averaged 7 percent annual growth for two years before he died in 1923 during the [[w:Roaring Twenties|Roaring Twenties]]. (2) [[w:Rutherford B Hayes|Rutherford B Hayes]], who took office in 1877 at the end of the [[w:Panic of 1873|Panic of 1873]], oversaw 5.5 percent growth per year for his 4 years as president. Those impressive sounding average growth figures are so short that they are not noticeably different from other minor features visible in Figure 1. 

FDR also dramatically reduced inequality. For over a century between about 1820 and 1932 the top 1 percent of the US population owned roughly 45 percent of net personal wealth in the US, while the bottom 90 percent owned roughly 22 percent. Shortly after FDR died, the wealth of the top 1 percent and bottom 90 percent were even at roughly 30 percent. 

FDR also oversaw dramatic growth in the power of organized labor. Their successes continued to reduce inequality to roughly 22 percent for the top 1 percent vs. 37 percent for the bottom 90 percent when Ronald Reagan became president. Since then, [[w:Reaganomics|Reaganomics]] have been rebuilding the [[w:Robber baron (industrialist)|Robber Baron]]s.&lt;ref&gt;This history of inequality is documented in [[:File:Share of US wealth 90p99.svg|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023]] in the Wikiversity article on &quot;[[w:Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]&quot; in Wikiversity, accessed 2026-10-13.&lt;/ref&gt;

FDR made enemies among the wealthiest and most powerful. In a [[q:Franklin D. Roosevelt#Address at Madison Square Garden (1936)|speech in Maddison Square Garden in 1936]], FDR mentioned, &quot;the old enemies of peace: business and financial monopoly, speculation, reckless banking, class antagonism, sectionalism, war profiteering. ... They are unanimous in their hate for me -- and I welcome their hatred.&quot; They doubtless could see that they could no longer afford as many personal servants -- maids, butlers, cooks, chauffeurs, etc. -- as they had before FDR. 

Broadcasting was relatively new when FDR took office. The economics of broadcasting naturally favored a very few networks, whose power concerned FDR. In 1941 his [[w:Federal Communications Commission| Federal Communications Commission]] (FCC) adopted the [[w:Mayflower doctrine|Mayflower doctrine]], which prohibited broadcasters from advocacy. When FDR died, people with power were able to get the Mayflower doctrine replaced by the more lenient [[w:Fairness Doctrine|Fairness Doctrine]], which opened the floodgates for entrepreneurs of hate, who drove the [[w:McCarthyism|McCarthy Red Scare]] in the US and many other problems since. Over time this became worse, facilitated by increasing concentration of ownership of the media. That contributed to the end of the Fairness Doctrine starting in 1985. That revocation and especially the more recent rise of for-profit social media have amplified the ability of entrepreneurs of hate to make money both for commercial social media and for themselves by disseminating lies about others, e.g., immigrants.&lt;ref&gt;On how for-profit social media companies promote entrepreneurs of hate, see, e.g., DiResta (2026) and the Wikiversity article on, &quot;[[Managing the good and the evils of social media]]&quot;, accessed 2026-10-12. DiResta (2026, p. 1) described the promotion of &quot;influencers&quot; by for-profit social media as essentially unleashed &quot;a war for power, influence, and reality itself—a conflict in which the ends justify any means, and one that has ripped America apart, threatened our very democracy, and consumed my own life for the past decade, leading to death threats, congressional subpoenas, lawsuits, and legal bills running into seven figures.&quot; In the interview on Wikiversity, DiResta said her legal bills to that point had exceeded $7 million.&lt;/ref&gt; 

And the rate of economic growth in the US (and the Netherlands and the UK) have been slowing in recent decades, as is visible in Figure 1. Inequality expert [[w:Thomas Piketty|Piketty]] (2022, p. 139) noted, {{quote| 

In the United States, the national income per inhabitant rose at a rate ... of 2.2% between 1950 to 1990 when the top tax rate reached on average 72%.  The top rate was then cut in half, with the announced objective of boosting growth.  But in fact, growth fell by half, to 1.1% per annum between 1990 and 2020 ... . [A]ll the data at our disposal today suggest that virtually confiscatory tax rates have been an immense historical success. ...[T]he battle for equality and education ... has made economic development and human progress possible, and not the veneration of property, stability, and inequality.}}

Table 1 compares Piketty's growth numbers with computations based on the data from both Maddison and [[w:MeasuringWorth|MeasuringWorth]]: The numbers are close but not exact. 

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;&quot;
|+ Table 1. US growth rate 1950 to 1990 to 2020
|-
! 
! colspan=3 | US GDP per capita !! colspan=2 | Growth (% / yr)
|- 
! source || 1950 !! 1990 !! 2020 !! 1950 - 1990 !! 1990 - 2020 
|-
| Piketty || - || - || - || 2.2% || 1.1%  
|- 
| Maddison || $15,240 || $36,982 || $54,379 || 2.24% || 1.29%
|- 
| MeasuringWorth || $16,208 || $40,191 || $61,127 || 2.30% || 1.41%
|}

Acemoglu and Robinson (2012, pp.  105-106) provide a historical analysis supporting Piketty's concern about inequality slowing growth. They note that in 1588, England, Spain and France {{quote|

were ruled by relatively absolutist monarchs: [[w:Elizabeth I|Elizabeth I]] in England, [[w:Philip II of Spain|Philip II]] in Spain, and [[w:Henry III of France|Henry III]] in France. All were battling with assemblies of citizens -- such as the [[w:Parliament of England#Tudor era (1485–1603)|Parliament]] in England, the [[w:Cortes Generales#Habsburg rule (16th–17th centuries)|Cortes]] in Spain, and the [[w:Estates General (France)#Revival in 1560–1614|Estates-General]] in France -- that were demanding more rights and control over the monarchy. ... [T]he English Parliament and the Spanish Cortes had power over taxation, while the Estates-General did not. In Spain this mattered little, because after 1492 the Spanish Crown had a vast American empire and benefited massively from the gold and silver found there. In England ... Elizabeth I was far less financially independent, so she had to beg Parliament for more taxes. In exchange, Parliament demanded concessions, in particular restrictions of the right of Elizabeth to create monopolies. ... In Spain the Cortes lost a similar conflict.}} 

Consistent with Acemoglu and Johnson (2023) and the discussion above, inequality tends to increase elite control of the media, which it makes it harder for commoners to get a fair hearing of their concerns. 

=== Media conflicts of interest in IP law ===

Major media have [[w:Conflict of interest|conflicts of interest]] in honestly reporting on anything involving [[w:Intellectual property|intellectual property]] (IP) law. An example of Cardwell's law, mentioned above, is the [[w:Copyright Term Extension Act|Sonny Bono Copyright Term Extension Act]], also known as the Mickey Mouse Protection Act of 1998. The copyright term on ''[[w:Steamboat Willie|Steamboat Willie]]'', the first [[w:Mickey Mouse|Mickey Mouse]] animated film that Disney distributed, was extended multiple times by the US Congress following extensive lobbying by the film industry led by [[w:The Walt Disney Company|the Walt Disney Company]]. ''Steamboat Willie'', which debuted in 1928, finally entered the [[w:Public domain in the United States|US public domain]] on January 1, 2024. 

Eric Eldred and others filed suit, claiming that allowing any number of retroactive extensions violated the [[w:Copyright Clause|Copyright Clause]] of the US Constitution, which says, {{quote|

[the United States Congress shall have power] To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.}}

In ''[[w:Eldred v. Ashcroft|Eldred v. Ashcroft]]'' (2003), the [[w:Supreme Court of the United States|US Supreme Court]]  disagreed with Eldred and allowed the statute to remain in effect. However, that case drew enough public attention that copyright terms have not (yet) been extended further, and ''Steamboat Willie'' is currently in the pubic domain. 

[[w:Lawrence Lessig|Lessig]] (2004) ''[[w:Free Culture (book)|Free Culture]]'' describes other examples, where US copyright law seems to ''limit'' rather than &quot;promote the Progress of Science and useful Arts&quot;. This includes a case where a 2002 freshman, Jesse Jordan, modified a search engine to make it easier for other student to find pictures, research, notes, movie clips, etc., on an [[w:intranet|intranet]] at [[w:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] (RPI). The [[w:Recording Industry Association of America|Recording Industry Association of America]] (RIAA) sued Jordan and three other students for close to $100 billion for copyright violations -- six times the total profit of the film industry in 2001. Jordan had an uncle, who was an attorney, who explained that the legal bills for fighting that case would be at least $250,000. That would bankrupt his parents; they could not recover that money, even if they won. Jordan had saved $12,000 to cover part of his college expenses. RIAA accepted that and settled the case.&lt;ref&gt;Lessig (2004, p. 42).&lt;/ref&gt; 

This is an example of a [[w:Strategic lawsuit against public participation|strategic lawsuit against public participation]] (SLAPP), which in this case rested on questionable interpretations of copyright law. The major media have conflicts of interest in honestly reporting on anything involving (a) copyright laws, because much of their income is derived from copyright royalties, or (b) anything that might offend their major accounts, including major corporations. And anti-SLAPP laws, which exist in some jurisdictions but not others,&lt;ref&gt;[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel for the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]], supports anti-SLAPP laws and the &quot;JAWBONE act&quot;, to limit the power of the government to threaten questionable actions, like the Trump administration's threats against the ABC network over a Jimmy Kimmel monolog. See the Wikiversity article on &quot;[[Foundation for Individual Rights and Expression]]&quot;, accessed 2026-10-16.&lt;/ref&gt; limit the ability of major corporations to abuse their power against potential challengers without such deep pockets.

Similarly Graves (2023) asked, &quot;Do copyrights and paywalls on refereed academic journals violate the US constitution?&quot; No one expects to make money from copyright royalties for drafts submitted to refereed academic journals. They do it out of a felt need (a) to contribute to the future of humanity or (b) to build their reputation or (c) because promotions at many universities are based partly on publication records. Graves is not alone in his concern about this: Many European countries do not allow reports of research they fund to be published behind paywalls.&lt;ref&gt;Dimi Zagorski is concerned that copyright exceptions for scientific research are different in different European countries, and that's a problem for the Wikimedia Foundation. See the discussion of this in the Wikiversity article on &quot;[[Wikimedia concerns with European copyright rules including AI and scientific research]]&quot;. That's the opposite of US law, which grants patents to universities, research institutions, private companies, and individuals for innovations developed under government contracts, per the Bayh-Dole Act of 1980, described by Baker (2016, p. 80).&lt;/ref&gt;

Regarding patent law, the major media have conflicts of interest in discussing proposed changes affecting major accounts, but they make little use of patent law themselves. This seems to have contributed to the claim that, &quot;[O]ver the last four decades ... stronger patent protection seems to be associated with slower growth&quot; in the US, according to [[w:Dean Baker|Baker]] (2016, pp. 77-78). A potentially similar phenomenon appears in a comparison of 21 different European countries in an analysis that displays &quot;a positive relationship between anticorruption measures in a country and rate of innovation in that country as measured by patents per capita&quot;, discussed by Aghion et al. (2021, Figure 15.1b, pp. 292-293).&lt;ref&gt;Usher and Kim-Leffingwell (2022) documented how each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN), relatively small nonprofits independent of major media organizations and not funded by advertising, in a US federal court district was associated with on average 1.4 additional prosecutions for political corruption per year between 2003 and 2019. They found on average 10 federal prosecutions for political corruption for each federal court district per year. Thus, 1.4 additional prosecutions per year per district represents an increase of 14 percent per member of INN.&lt;/ref&gt;

=== Immigrants === 

We now return to discussing the initiation of economic growth following the beheading of Charles I in 1649 visible in Figure 1. Charles I was replaced by [[w:English Council of State|Lord Protector Cromwell]], who refused to authorize as many monopolies as his predecessors, as mentioned above. He also encouraged Jews and dissenting Christians to settle in England, as mentioned by Mokyr (2017, p. 53), cited above. 

But Mokyr is not alone in claiming that immigrants contribute to economic growth. Aghion et al. (2021, p. 266) cite Arkolakis et al. (2020) for explaining how the period 1880–1920 is noteworthy for two parallel evolutions. First, the United States caught up with England and France to become the most technologically advanced country in the world as well as the wealthiest in terms of GDP per capita. Second, the United States welcomed large numbers of immigrants, most notably from Europe. The purpose of the cited study was to demonstrate a causal connection between these two phenomena: in other words, to show that European immigration was responsible for the United States’ taking the lead in technology. ... More precisely, if the United States had completely closed its borders to all foreign immigrants in 1880, its per capita GDP would have been 30 percent lower than the level actually attained in 1920&quot;. 

=== Industrial Revolution ===

There seems to be a broad consensus that the [[w:Industrial Revolution|Industrial Revolution]] began in England. However, there is less agreement on when it began. 

Mokyr (2017, p.  3) says, &quot;The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society.&quot; He described &quot;three cultural elements&quot; required for &quot;sustainable economic growth propelled by technological progress&quot;: (1, 2) The belief that it was both ''possible'' and ''desirable''. (3) Supportive policy measures had to be implemented. &quot;[B]efore 1750 they did not produce anything like an Industrial Revolution or sustainable economic growth propelled by technological progress.&lt;ref&gt;Mokyr (2017, pp. 19-20).&lt;/ref&gt; Acemoglu (2026, p. 53) similarly mentioned the eighteenth century in saying, &quot;The breakthroughs in production technology that started in the middle of the eighteenth century in Britain unleashed the Industrial Revolution.&quot; 

Aghion et al. (2021, esp. pp. 29-30) offer a ''later'' date for the start of the Industrial Revolution: They report that, &quot;European economies prior to 1820 experienced a series of phases of growth and decline. This raises a question: How did the pre-1820 period differ from the takeoff that started with the first Industrial Revolution?&quot; Aghion et al. (2021, p. 20) describe the &quot;first Industrial Revolution&quot; as having been triggered by &quot;the invention of the steam engine by James Watt in the 1770s ... . The invention of electricity set off the second Industrial Revolution, whose golden age was in the first half of the twentieth century. This second revolution originated with the invention of the [[w:Incandescent light bulb|light bulb]] by [[w:Thomas Edison|Thomas Edison]] in 1879 and of the [[w:dynamo|dynamo]] by [[w:Werner von Siemens|Werner von Siemens]] in 1866.&quot;&lt;ref&gt;The dynamo was independently invented by [[w:Henry Wilde (engineer)|Henry Wilde]], [[w:Charles Wheatstone|Charles Wheatstone]], [[w:Samuel Alfred Varley|Samuel Alfred Varley]] and Siemens, all about the same time, according to the Wikipedia article on [[w:Dynamo#Practical designs|dynamo]], accessed 2026-10-18.&lt;/ref&gt;

The image in Figure 1 raises questions about these claims by Mokyr, Acemogu, and Aghion et al.: It shows growth followed by stagnation in Holland but not &quot;phases of growth and decline&quot; with the exception of wars and [[w:Herbert Hoover|Herbert Hoover]]. It also does ''not'' show the slope (i.e., rate of growth) increasing starting around 1750 or the 1770s or 1879 or 1886. Instead it shows sustained growth in GDP per capita for England / Great Britain / the UK starting ''abruptly'' after 1649 with economic disruptions during the reign of [[w:Anne, Queen of Great Britain|Queen Anne]] (1702-1714) and the [[w:War of the Spanish Succession|War of the Spanish Succession]] (1701-1714). This was followed by relatively steady but slower growth as England was ruled by Kings [[w:House of Hanover#Monarchs of Great Britain, Ireland, and Hanover|George I, II, and III]] (1714-1820). Another dramatic increase in the slope started abruptly about 1819. 

Acemoglu and Johnson (2023, chapter 1) are more consistent with Figure 1. They said that, &quot;the Industrial Revolution ... [was] rooted in institutional changes that started empowering the middling sort of English people from the sixteenth and seventeenth centuries onward.&quot; ... However, &quot;the quality of life of a European peasant circa 1700 was not much different from that of an Egyptian peasant two thousand or seven thousand years previously.&quot; (chapter 5) They also claim that, &quot;English national production per capita doubled from 1500 to 1700&quot;, (chapter 5), and &quot;Growth emerged slowly in the second half of the eighteenth century as the Industrial Revolution, based on major technological breakthroughs and their application in industry, took root.&quot; (p. 45, chapter 2)

Indeed, Figure 1 shows English GDP per capita doubling between 1500 and 1700, but it appears to have been fairly constant from 1500 to 1649, then roughly doubled in the next half century. [The data on England up to 1700 in Figure 1 came from Broadberry et al. (2015), which was not available when Acemoglu and Robinson were writing ''Why Nations Fail'', which appeared in 2012.] 

Acemoglu and Robinson (2012, ch. 11) describe the financial instability and slow growth during the reigns of Queen Anne and George I, II, and III. After the Glorious Revolution, France, Spain, the pope, and others refused to recognize [[w:William III of England|William and Mary]] (1689-1702) and then Queen Anne as heads of state of England. Instead they continued to support the claims of James II and then his son, [[w:James Francis Edward Stuart|James Francis Edward Stuart]] (the &quot;Old Pretender&quot;, a brother of Queen Anne), after James II died in 1701. During the War of the Spanish Succession, each side tried to exploit the internal divisions of the other. Owen (1938, ch. 8) said France almost succeeded in an attempt in 1708 to [[w:Planned French invasion of Britain (1708)|land between 5 and 6 thousand soldiers in northeast Scotland]]. [[w:Louis XIV|Louis XIV]] had hoped to get the Old Protector named as King James VIII of Scotland. There were still [[w:Jacobitism|Jacobites]] in Scotland, who supported the Old Pretender and likely would have supported the invaders if they had landed. If they had succeeded, it would have created major problems for Queen Anne and her supporters. However, the Royal Navy prevented the invaders from disembarking, and they returned to France. 

When Queen Anne died in 1702, she was succeeded by George I. His Parliament was dominated by the Whig party, which represented the new mercantile and economic interests and had been the main organization behind the Glorious Revolution. Once in power, they were tempted to use their newly found position to prey on the rights of others but were constrained both by competing groups in Parliament, &quot;and by the very institutions that they had fought to introduce to strengthen Parliament and to prevent the emergence of a new absolutism and the return of the Stuarts&quot;,&lt;ref&gt;After James II fled England, the new government worked to block [[w:Jacobitism|Jacobitism]], which attempted to restore the [[w:House of Stuart|House of Stuart]] to the thrones of Scotland and Britain. When James II died, his son became titular head of this movement, who became known as &quot;the [[w:James Francis Edward Stuart|Old Pretender]]&quot;. After he died, he was replaced by his son, who became known as &quot;the [[w:Charles Edward Stuart|Young Pretender]]&quot;.&lt;/ref&gt; according to Acemoglu and Robinson (2012, pp. 302-303). The Whigs gave &quot;[[w:Common land|common land]]&quot; to nobles, confiscating it from commoners. Some commoners, who had traditionally shared that land, joined &quot;Blacks&quot;, so called, because they blackened their faces when hunting on former commons or destroying the property of the nobility or extorting money from nobles or their employees. Parliament retaliated by passing the [[w:Black Act 1723|Black Act 1723]], under which someone accused of [[w:Poaching|poaching]] a [[w:Hare|hare]], for example, could under certain circumstances receive a summary judgment of guilty and then be executed without delay if apprehended later.&lt;ref&gt;Newman and Brown, eds. (1997, p. 58). See also the Wikipedia article on &quot;[[w:Black Act 1723|Black Act 1723]]&quot;, accessed 2026-10-17.&lt;/ref&gt; 

But not all accused were summarily executed. In 1724 &quot;John Huntridge was accused of aiding deer stealers and abetting known Blacks, both crimes punishable by hanging. The prosecution of Huntridge came right from the top, initiated by the Regency Council of Lords Justices ... . After a trial of eight or nine hours, the jury found Huntridge innocent, partly on procedural grounds, since there were irregularities with the way the evidence had been collected. ... The jury was made up not of Huntridge’s peers, but of major landowners and gentry, who ought to have sympathized&quot; with the noble, whose property had been attacked by Blacks allegedly aided by Huntridge.&lt;ref&gt;Acemoglu and Robinson (2012, pp. 304-305).&lt;/ref&gt; 

These kinds of actions by the Parliaments of George I, II, and III might explain the slower growth during that period compared to that of the last half of the seventeenth century. That slower growth ended abruptly at the time of the [[w:Peterloo massacre|Peterloo Massacre]], 1819-08-16, visible in Figure 1. The growth of factories implied growth of cities, where workers lived closer together and could more easily organize. A campaign arose with multiple demands: They opposed the [[w:Corn Laws|Corn Laws]], which kept prices high for cereal grains and products made from them. They also demanded &quot;universal [adult male] suffrage&quot; and secret ballots in elections. A series of public meetings discussing these grievances led to one 1819-08-16, which attracted roughly 60,000. Acemoglu and Robinson (2012, p. 207) reported that, {{quote|

The authorities were very nervous about the meeting, and a force of six hundred cavalry of the [[w:15th The King's Hussars|Fifteenth Hussars]] had been assembled. As the speeches began, a local magistrate decided to issue a warrant for the arrest of the speakers. As police tried to enforce the warrant, they met with the opposition of the crowd, and fighting broke out. At this point the Hussars charged the crowd. Within a few chaotic minutes, eleven people were dead and probably six hundred wounded. The ''[[w:Manchester Observer|Manchester Observer]]'' called it the Peterloo Massacre. But given the changes that had already taken place in economic and political institutions, long-run repression was not a solution in England. The Peterloo Massacre would remain an isolated incident. Following the riot, the political institutions in England gave way to the pressure, and the destabilizing threat of much wider social unrest, particularly after the 1830 revolution in France against [[w:Charles X of France|Charles X]], who had tried to restore the absolutism destroyed by the French Revolution of 1789. In 1832 the government passed the [[w:Reform Act 1832|First Reform Act]]. It enfranchised [[w:Birmingham|Birmingham]], [[w:Leeds|Leeds]], [[w:Manchester|Manchester]], and [[w:Sheffield|Sheffield]], and broadened the base of voting so that manufacturers could be represented in Parliament. ... [I]n 1846 they managed to get the hated Corn Laws repealed ... .}}

The ''Manchester Observer'' closed roughly two years later after [[w:Manchester Observer#Closure by repeated prosecution|repeated prosecutions]] for [[w:Seditious libel|seditious libel]]. However, the British government placed enough other limits on its power that enough other commoners were allowed to become successful entrepreneurs that the rate of economic growth in the UK increased immediately, as indicated by the increase in the slope of the UK line in Figure 1 until World Wars I and II. 

== In sum ==

The above suggests that the rate of economic growth changes with either (a) a change in administration -- a new head of state -- or (b) some other major event like war, the Peterloo Massacre, or the [[w:Wall Street crash of 1929|Wall Street crash of 1929]]. Policy changes that can increase growth include (1) limits on grants of monopolies, (2) diverse ownership and control of the major media and (3) support for immigrants. 

However, correlation does not imply causation. [[w:Randomized controlled trial|Randomized controlled trial]]s have made important contributions to human understanding of economic growth, as documented by Banerjee, Duflo, and Kramer, cited above and recommended in the Wikiversity article on &quot;[[Information is a public good: Designing experiments to improve government]]&quot;. Failing that, more analyses of data from different political jurisdictions with different media policies could be useful, as described by Neff and Pickard (2024).&lt;ref&gt;Graves (2025) includes an analysis of the data in Neff and Pickard (2024).&lt;/ref&gt; 

== Caveats ==

We should be more confident of the results of randomized controlled trials, as recommended by Banerjee, Duflo, and Kremer, than of anyone interpreting the historical record, even by Nobel economists. 

The above attempts to accurately reflect the perspectives of all sources cited. Readers are invited to revise this article, subject to the standard Wikimedia rules of writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”{{'}}''' page]] -- and treating others with respect. 

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!-- Daron Acemoglu (2026-08-11) What Happened to Liberal Democracy?: Remaking a Politics of Shared Prosperity (Function)--&gt;{{cite Q|Q141241291}}
* &lt;!--Daron Acemoglu and Simon Johnson (2023) Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity (PublicAffairs)--&gt;{{cite Q|Q125292212}}
* &lt;!--Daron Acemoglu and James A. Robinson (2012) Why Nations Fail (Crown)--&gt;{{cite Q|Q7997840}} 
* &lt;!--Philippe Aghion, Céline Antonin, and Simon Bunel (2021) The Power of Creative Destruction: Economic Upheaval and the Wealth of Nations (Harvard University Press)--&gt;{{cite Q|Q137641358}}
* &lt;!--Ajzenman, Nicolas (29 January 2023). &quot;Migrants don't cause crime rates to increase — but false perceptions endure anyway&quot;. The Conversation.--&gt;{{cite Q|Q141758664}}
* &lt;!--Samuel Alexander (2026-07) &quot;Game Theory and the Dilemma of Degrowth: Why Embrace Degrowth if Other States or Societies Do Not?&quot;--&gt;{{cite Q|Q141968019}}
* &lt;!--Costas Arkolakis, Michael Peters, and Sun Kyoung Lee, (2020) &quot;European Immigrants and the United States’ Rise to the Technological Frontier&quot;--&gt;{{cite Q|Q141678287}}
* &lt;!--Dean Baker (2016) Rigged: How globalization and the rules to the modern economy were structured to make the rich richer--&gt;{{cite Q|Q100216001}}
* &lt;!--Abhijit V. Banerjee and Esther Duflo (2019) Good Economics for Hard Times (PublicAffairs)--&gt;{{cite Q|Q85764011}}
* &lt;!--Jutta Bolt and Jan Luiten van Zanden (2024-09-18) &quot;Maddison style estimates of the evolution of the world economy: A new 2023 update&quot;, Journal of Economic Surveys, 1-41--&gt;{{cite Q|Q126723821}}
* &lt;!--S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)--&gt;{{cite Q|Q57945943}}
* &lt;!--Donald S. L. Cardwell (1972) Turning Points in Western Technology (Neale Watson, Science History Pubs)--&gt;{{cite Q|Q141795083}}
* &lt;!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition--&gt;{{cite Q|Q141156783}}
* &lt;!--Renée DiResta and Rachel Kleinfeld (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies--&gt;{{cite Q|Q141158362}}
* &lt;!--Katarzyna Dośpiał-Borysiak (2017) &quot;Model of state management of petroleum sector -- Case of Norway--&gt;{{cite Q|Q141774179}}
* &lt;!--Rachel Foxley (2013) The Levellers: Radical Political Thought in the English Revolution (Oxford University Press)--&gt;{{cite Q|Q136227988}}
* &lt;!--Benjamin Gonzalez O’Brien, Loren Collingwood, Stephen Omar El-Khatib (2017-05-07) &quot;The Politics of Refuge: Sanctuary Cities, Crime, and Undocumented Immigration--&gt;{{cite Q|Q55932800}}
* &lt;!--Graves (2025-01-08) Defend Free Speech For All (Town Hall)--&gt;{{cite Q|Q132855526}}
* &lt;!--Graves (2023-03-21)  Do copyrights and paywalls on academic journals violate the US Constitution?--&gt;{{cite Q|Q126116187}}
* &lt;!--Christopher Hibbert, Ben Weinreb, John Keay, Julia Keay, and Matthew Weinreb (2011-09-09) The London Encyclopaedia, 3rd ed.--&gt;{{cite Q|Q137188993}}
* &lt;!-- Historian, USPS (2010) Postage Rates for Periodicals --&gt;{{cite Q|Q112167199}}
* &lt;!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse--&gt;{{cite Q|Q54641943}}
* &lt;!--Volker Leppin and Timothy J. Wengert (2015) &quot;Sources for and against the Posting of the Ninety-Five Theses&quot;--&gt;{{cite Q|Q141737347}}
* &lt;!--Lawrence Lessig (2004) Free Culture--&gt;{{cite Q|Q830526}}
* &lt;!--Olivier Marie and Paolo Pinotti (2024-02-01) &quot;Immigration and Crime: An International Perspective&quot;, Journal of Economic Perspectives. 38 (1): 181–200--&gt;{{cite Q|Q128434874}}
* &lt;!--Martínez, Daniel E.; Martínez-Schuldt, Ricardo D.; Cantor, Guillermo (2017). &quot;
Providing Sanctuary or Fostering Crime? A Review of the Research on 'Sanctuary Cities' and Crime&quot;
. Sociology Compass. 12 e12547. doi:10.1111/soc4.12547. ISSN 1751-9020.--&gt;{{cite Q|Q55932802}}
* &lt;!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) --&gt;{{cite Q|Q104888067}}.
* &lt;!-- Robert W. McChesney; John Nichols (2016). People get ready--&gt;{{cite Q|Q87619174}}
* &lt;!--Robert Michels (1911)  Les Partis politiques, essai sur les tendances oligarchiques des démocraties--&gt;{{cite Q|Q7210216}}
* &lt;!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Princeton University Press)--&gt;{{cite Q|Q136118590}}
* &lt;!--Joel Mokyr (1994) &quot;Cardwell's Law and the political economy of technological progress&quot;, Research Policy 23(5)561-574--&gt;{{cite Q|Q139772607}}
* &lt;!--Ashley N. Muchow &amp; Taisiia Stanishevska (2026) &quot;From Headlines to Handcuffs: Local News Coverage of Immigrant Crime and Hispanic Arrest&quot;, Justice Quarterly, vol. 43(3) (https://doi.org/10.1080/07418825.2025.2550307)--&gt;{{cite Q|Q141760694}}
* &lt;!--Neff and Pickard (2024) &quot;Funding Democracy: Public Media and Democratic Health in 33 Countries&quot;--&gt;{{cite Q|Q131468289}}
* &lt;!--Gerald G. Newman and Leslie Ellen Brown, eds. (1997) Britain in the Hanoverian Age, 1714–1837: An Encyclopedia--&gt;{{cite Q|Q141759196|authors = Gerald G. Newman and Leslie Ellen Brown, eds.}}
* &lt;!--John Hely Owen (1938)  War at Sea Under Queen Anne 1702–1708--&gt;{{cite Q|Q141757101}}
* &lt;!--John Perkins (2004) Confessions of an Economic Hit Man--&gt;{{cite Q|Q400098}}
* &lt;!--Andrew Pettegree (2015-10-27) Brand Luther: How an Unheralded Monk Turned His Small Town into a Center of Publishing, Made Himself the Most Famous Man in Europe--and Started the Protestant Reformation--&gt;{{cite Q|Q141761223}}
* &lt;!--Thomas Piketty (2022) A Brief History of Equality--&gt;{{cite Q|Q139886018}}
* &lt;!--Joad Raymond (2006-11-02) Pamphlets and Pamphleteering in Early Modern Britain--&gt;{{cite Q|Q141761849}}
* &lt;!--James A. Robinson (2025-04) &quot;Paths to the periphery&quot;, NBER Working Paper 33671--&gt;{{cite Q|Q141967690}}
* &lt;!--Nick Roll (2017-07-16) &quot;Correcting Jeff Sessions: Attorney general railed against crime in sanctuary cities, citing two professors’ work. But his claims are contrary to their conclusions.--&gt;{{cite Q|Q141735231}}
* &lt;!--J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)--&gt;{{cite Q|Q137669960}}
* &lt;!--I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)--&gt;{{cite Q|Q137669987}}
* &lt;!--Kelly Swanson (2017-09-17) &quot;The Trump administration rejects its own study finding refugees help the economy--&gt;{{cite Q|Q141900700}}
* &lt;!--Ted Trainer (2025-03) &quot;The 2024 Nobel Prize for Economics: An example of how conventional economics misrepresents reality--&gt;{{cite Q|Q142065447}}
* &lt;!--Union of Concerned Scientists (2019-01-11) &quot;The Trump Administration Manipulates Study on Economic Benefits of Refugees in US&quot;--&gt;{{cite Q|Q141898260|author=Union of Concerned Scientists}}
* &lt;!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption --&gt;{{Cite Q|Q134715465}}
* &lt;!--J. L. van Zanden  and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30--&gt;{{cite Q|Q137670058}} 
* &lt;!--Brittney Washington (2018-06-19) &quot; A Pamphlet War in England, 1641-1643&quot;--&gt;{{cite Q|Q141822147}}
* &lt;!--Tom K. Wong (2017-01-26) &quot;The Effects of Sanctuary Policies on Crime and the Economy&quot;--&gt;{{cite Q|Q98775875}}

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::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]''

== Abstract ==

This article first identifies and plots GDP per capita for leading countries in the Maddison Project data then discusses comments by Nobel prize winning economists and others to interpret what appears in said plot. No one really knows how to make economies grow. However, growth at the technology frontier requires creative destruction, and immigrants tend to be more entrepreneurial than native born. Also, creative destruction is facilitated and inequality is limited by diversity of ownership and control of the major media. 

== Maddison Historical Statistics == 

[[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD = initially [[w:Holland|Holland]] to 1807 and the [[w:Netherlands|Netherlands]] after; GBR = initially England to 1700, Great Britain to 1870, then the UK; USA starting in 1650 with the 13 colonies that became the US; SGP = Singapore starting in 1870).&lt;ref&gt;See Bolt and van Zanden (2024) for the Maddison Data generally, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, Sugimoto (2011) for Singapore to 2007, van Zanden and van Leeuwen (2012) for the data on Holland 1349–1807, and Smits et al. (2000) for the data on the Netherlands 1808-1913.&lt;/ref&gt; The dramatic drop in the orange line between 1807 and 1808 is due to the change in reference from Holland to the Netherlands; at that time Holland was the most prosperous region of the Netherlands.&lt;ref name=vanZ_B&gt;Private email exchange with van Zanden and Bolt, 2026-01-08.&lt;/ref&gt;'']]

The [[w:Maddison Project|Maddison Historical Statistics Project]] (aka Maddison Project) is widely recognized as the premier data source on historical macroeconomic data. It is named after [[w:Angus Maddison|Angus Maddison]] (1926-2010), who was a pioneer in [[w:National accounts|national accounts]], especially estimating [[w:Gros Domestic Product|gross domestic product]] (GDP) per capita (GDPpc) and population (pop) at various times dating back to the year 1 of the [[w:Common Era|Common Era]] (CE / AD). After Maddison's death, the Maddison Project has continued to update the database to include the best sources available on any country for any historical period since year 1. 

When accessed 2026-10-10 the version of the data available had been released 2024-09-18 containing data to 2022.&lt;ref&gt;Bolt and van Zanden (2024).&lt;/ref&gt; This dataset includes 23,280 observations on at least one of GDPpc and pop for an average of 137 years for 169 countries. Only 18 countries had the highest GDPpc for at least one year. Annual data starts in 1252, and that's only for one country, GBR = England / Great Britain / the United Kingdom. NLD (Holland / the Netherlands) had the highest GDPpc for 97 percent of the years between 1349 and 1807. The drop in the orange line in Figure 1 between 1807 and 1808 is due to the change of reference from Holland to the Netherlands.&lt;ref name=vanZ_B/&gt; Visual comparison of the orange and red lines in Figure 1 suggests that the GDPpc of Great Britain overtook that of the Netherlands a few decades earlier and overtook Holland a few decades later. This analysis focuses on data since 1349, in part because data were available for at most 4 countries for most years prior to that date. (The 3-letter country codes used here are [[w:ISO 3166-1 alpha-3|ISO 3166-1 alpha-3]], used in the Maddison data.) 

We are interested in technology leaders, so we eliminate the [[w:petrostates|petrostates]] of Qatar (QAT), Kuwait (KWT), and United Arab Emirates (ARE), plus Norway (NOR); the latter owes a substantial portion of their wealth to democratic management of [[w:North Sea oil|North Sea oil]].&lt;ref&gt;Dośpiał-Borysiak (2017).&lt;/ref&gt; With these restrictions 12 countries had the highest GDPpc for at least one year. However, FRA (France), ITA (Italy), SWE (Sweden), BEL (Belgium), NZL (New Zealand) and CHE (Switzerland) led for at most 7 of the 674 years between 1349 and 2022; we eliminate those, because including them would seem to add more clutter than clarity. We also eliminate AUS (Australia), which led for 17 of the 39 years between 1853 and 1891, when it was competing for the leadership with GBR and USA, for the same reason. And we eliminate LUX (Luxembourg), which has a population under a million. 

This leaves NLD, GBR, USA, and SGP (Singapore), plotted in Figure 1; Singapore has led since 2010.&lt;ref&gt;More details of this analysis are available in the chapter on &quot;[[Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs|The impact of the media on political economy since the time of the Pharaohs]]&quot;&quot; in the book-in-progress on ''[[Media Literacy and You]]'' on Wikiversity.&lt;/ref&gt; This plot has some interesting features, which we discuss in the remainder of this research report. 

== Nobel economists and others ==

[[w:Abhijit Banerjee|Banerjee]] and [[w:Esther Duflo|Duflo]] (2019, p. 207) said, &quot;despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive.&quot; That appeared shortly before they shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]] with [[w:Michael Kremer|Michael Kremer]] for documenting how [[w:Randomized controlled trial|randomized controlled trial]]s could be used to alleviating poverty in [[w:Development economics|development economics]].&lt;ref&gt;For the present discussion, we assume that enough [[w:Externality|externalities]] have been sufficiently internalized with appropriate [[w:Pigouvian tax|Pigouvian tax]]es to make growth desirable. This converts discussions of [[w:Degrowth|degrowth]], e.g., by Alexander (2026), into evaluations of whether externalities have actually been sufficiently internalized and what to do if they are not. Internalizing externalities appropriately will not be easy as long as those who benefit from not paying for damages they cause to customers, employees and neighbors control substantial portions of the funds for the media. However, for the present discussion, we assume this is not an issue.&lt;/ref&gt; 

While Banerjee and Duflo insisted that no one knows how to make an economy grow, other leading economists have recommended partial solutions. For example, [[w:Joel Mokyr|Mokyr]] (2017, p. 53) said, &quot;The willingness of the Netherlands and later Britain and the United States to tolerate Jews and dissenting Christians contributed a great deal to their economies, especially in high-skilled manufacturing and financial services.&quot; In other words, immigrants made an outsized contribution to the growth of NLD, GBR, and US, visible in Figure 1. Mokyr's comment appeared over a year before Banerjee and Duflo (2019) and roughy eight years before Mokyr shared the 2025 Nobel Memorial Prize in Economics with [[w:Philippe Aghion|Aghion]] and [[w:Peter Howitt (economist)|Howitt]]. 

Mokyr (2017, p. 255) also said that, &quot;intellectual innovation will eventually end up being increasingly resisted by an incumbent technology; if that resistance is not overcome, technological progress may be extinguished.&quot; That resistance, he suggested, drives {{quote|

the empirical regularity that no society remains at the cutting edge of technological creativity for very long&lt;ref&gt;Mokyr (2017, p. 260).&lt;/ref&gt;.}}

Mokyr (2017, p. 260; 1994) called this &quot;Cardwell's Law&quot; after Cardwell (1972), which describes {{quote|

the interaction between technological creativity and political competition ... based on the observation that technological progress encounters resistance from various groups that believe they stand to lose from innovation. These pressure groups will try to manipulate the political system to suppress successful innovation.}}

[[w:Robert Michels|Michels]] (1911) called this &quot;the [[w:Iron law of oligarchy|iron law of oligarchy]]&quot;. Further documentation of this appears in Acemoglu and Robinson (2012) and Acemoglu and Johnson (2023), who shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]]. 

Figure 1 begins with growth in England and Holland between 1349 and 1400, likely due to the fact that the [[w:Black Death|Black Death]] (1346-1353) killed between 30 and 60 percent of the population: Since land was not destroyed, survivors were likely able to produce more agricultural products for the same labor. Also, under the feudal system in place before the Black Death, commoners were not allowed to move without the permission of their employer, the Lord of the land where they worked. Efforts by elites to prevent commoners from moving and obtaining higher wages failed in England.&lt;ref&gt;Acemoglu and Robinson (2012, esp. pp. 99-101) note that surviving commoners in England were able to bargain for higher wages. The authorities passed emergency legislation in 1349 and 1351 that made it a crime to offer or accept wages higher than before the plague or to refuse to work or break an existing contract. In 1361 the penalties were increased to include imprisonment and even branding. This led to [[w:Peasants' Revolt|a major insurrection in 1381]] led by [[w:Wat Tyler|Wat Tyler]], which was put down with over 1,300 deaths. After that insurrection, authorities in England allowed commoners to demand and accept higher wages. Commoners were also allowed to move without the permission of their Lord, which had ''not'' been accepted before. By contrast, the treatment of commoners in Russia was reportedly ''worse'' than before the plague. Changes in the treatment of commoners between England and Russia seemed to vary depending on proximity to these extremes, with Austria more like Russia and France more like England.&lt;/ref&gt; The growth in Holland from 1492 has been has been attributed to [[w:Expulsion of Jews from Spain|Jews expelled from Spain settling]] in Holland and starting new businesses. 

The iron law of oligarchy provides an explanation for why growth stopped in England around 1400, Holland around 1600, and why growth has been slowing in recent decades in all the countries plotted in in Figure 1. It can also explain [[w:History of the Roman Empire|the rise and fall of the Roman Empire]], of the [[w:Maya civilization|Maya civilization]], and other societies that became very rich from allowing commoners to become entrepreneurs and then declined after previous winners changed the rules of the political economy to block further [[w:creative destruction|creative destruction]]. This includes how [[w:History of the Republic of Venice|Venice]] became &quot;possibly the richest place in the world, with the most advanced set of inclusive economic institutions&quot;, dominating trade in the [[w:Mediterranean Sea|Mediterranean]] based on a system that encouraged entrepreneurial commoners to undertake risky trading ventures starting after 810 CE and ending when leaders made wealthy by such ventures changed the rules to end that system starting in 1297, destroying their dominance of Mediterranean trade &quot;and converting Venice into a museum&quot;, in the words of Acemoglu and Robinson (2012, pp. 152-155).

Acemoglu and Robinson (2012) suggest three primary distinctions between the poorest and wealthiest nations. The poorest tend to be [[w:failed state|failed states]] like [[w:Somalia|Somalia]], which are poor, because of a substantial lack of rule of law and continued violent conflict between warring groups. Acemoglu and Robinson divide other nation states between &quot;extractive&quot; and &quot;inclusive&quot; regimes. &quot;Extractive&quot; regimes tend to be poor, because elites write the rules to protect their interests against the competition that generates the creative destruction required for economic growth at the technology frontier. &quot;Inclusive&quot; regimes limit monopolies and encourage enough commoners to become entrepreneurs that over time the economy tends to produce more with less, increasing the wealth of the nation. 

[[w:Ted Trainer|Trainer]] (2025) described the 2024 Nobel Prize for Economics as, &quot;An example of how conventional economics misrepresents reality&quot;, blaming the poor for their poverty, e.g., in failing to cite [[w:John Perkins (author)|Perkins]] (2004) ''[[w:Confessions of an Economic Hit Man|Confessions of an Economic Hit Man]]''. Doth professor Trainer protest too much? Chapter 9 in Acemoglu and Robinson (2012) describes, &quot;Reversing Development: How European colonialism impoverished large parts of the world&quot;, which describes several historical cases where a colonial power subordinated a local extractive elite, increasing the severity of their extractive policies, further impoverishing the vast majority of that locality. The [[w:Dutch East India Company|Dutch East India Company]] did this to most of [[w:Indonesia|Indonesia]], but they could not do this in the [[w:Banda Islands|Banda Islands]], which lacked a central authority whom the Dutch could coerce into signing a monopoly treaty. Politics there were more inclusive, with many small autonomous city-states, run by village meetings of citizens. To avoid having to compete with English, Portuguese, Indian, and Chinese merchants, the Dutch massacred &quot;almost the entire population of the islands, probably about fifteen thousand people. ... [O]nly a few were left alive, enough to preserve the know-how necessary for mace and nutmeg production. After this genocide was complete,&quot; the Dutch repopulated the Banda islands with a hierarchical plantation society.&lt;ref&gt;Acemoglu and Robinson (2012, p. 248).&lt;/ref&gt; 

[[w:Botswana|Botswana]] was one of the few places in the colonized world that successfully avoided this fate. In 1895 chiefs of three of the eight [[w:Tswana people|Tswana states]] in what is now Boswana sailed to England with a charm offensive to save their country from [[w:Cecil Rhodes|Cecil Rhodes]]. They met with [[w:Joseph Chamberlain|Joseph Chamberlain]], then colonial secretary, before and after an extensive speaking tour of England. With the favorable coverage that the chiefs got in the English press, Rhodes got a railroad through their country and nothing more. When Botswana gained independence in 1966, they had relatively inclusive political institutions that allowed them to grow economically, escaping the fate of Black Africans in [[w:Rhodesia|Rhodesia]] / [[w:Zimbabwe|Zimbabwe]] to their north and [[w:South Africa|South Africa]] to their south.&lt;ref&gt;Acemoglu and Robinsion (2012, pp. 404-405).&lt;/ref&gt; 

Robinson (2025) provided a related explanation for the poverty of Zimbabwe, whose 2026 population is roughly 17 million with a 2025 GDP per capital of roughly $5,400 at PPP. '[D]uring fieldwork in Zimbabwe in 2017 I had breakfast with the CEO of a subsidiary of a large multinational company. I was trying to understand the political incentives and mechanisms that had led to extractive economic institutions in the country, leading it to be substantially poorer today than it was at the time of independence in 1980. ... At some point in the conversation the CEO looked up at me and said: “I really don’t know what your problem is, it really isn’t complicated, what is it that you can’t understand? ... [T]here’s plenty of money in this country for 300 people.”'

Mokyr (2017, p. 60) also mentioned &quot;entrepreneurs of hate&quot;, who convince others to dislike some group in a way that may benefit those &quot;entrepreneurs&quot; and major media organizations, who promote such &quot;entrepreneurs&quot; to please those who control much of the money for the media. For-profit social media have been particularly effective in herding their audiences into echo chambers, reinforcing their preconceptions in ways that increase political polarization and violence and threaten democracy.&lt;ref&gt;See, e.g., DiResta (2026), DiResta and Keinfeld (2025), and the Wikiversity article on &quot;[[Managing the good and the evils of social media]]&quot;, accessed 2026-10-14.&lt;/ref&gt; When successful this protects current elites from competition that would otherwise drive [[w:creative destruction|creative destruction]], benefitting the public at large while threatening existing elites.  

For example, most research on immigrants suggest that on average immigrants are more entrepreneurial and no more likely to commit crime than native born,&lt;ref&gt;e.g., Martínez et al. (2017); Muchow and Stanishevska (2026); Ajzenman(2023).&lt;/ref&gt; and some studies report that [[w:Sanctuary city|sanctuary jurisdiction]]s have on average higher median incomes and less crime.&lt;ref&gt;Martínez et al. (2017); Wong (2017). See also the Wikipedia article on &quot;[[w:Sanctuary city|sanctuary city]]&quot;, accessed 2026-10-12.&lt;/ref&gt; Where immigrant crime is higher, it typically result from anti-immigrant labor market restrictions or from young less-educated males with irregular status.&lt;ref&gt;Marie and Paolo (2024). See also the Wikipedia article on &quot;[[w:Immigration and crime|Immigration and crime]]&quot;, accessed 2026-10-12.&lt;/ref&gt; 

But major media outlets make money from featuring charismatic figures, who attract an audience without offending people who control much of the money for the media. In 2017 President [[w:Immigration policy of the first Trump administration|Trump]]'s Attorney General [[w:Jeff Sessions|Jeff Sessions]] quoted a research study that supposedly showed that sanctuary cities had more crime than cities without sanctuary policies. The authors of that study insisted that Sessions had ''misrepresented'' their work, which concluded &quot;that sanctuary policies have no effect on crime rates, despite narratives to the contrary.&quot;&lt;ref&gt;Roll (2017); Gonzalez O’Brien, et al. (2017).&lt;/ref&gt; Similarly, a 2017-09-19 report described how, &quot;The Trump administration rejects its own study finding refugees help the economy&quot;. &lt;ref&gt;Swanson (2017).&lt;/ref&gt; Just under three months later, the [[w:Union of Concerned Scientists|Union of Concerned Scientists]] (UCS) described how, &quot;The Trump Administration Manipulates Study on Economic Benefits of Refugees in US&quot;: The [[w:Refugee Act of 1980|Refugee Act of 1980]]  requires the president to set by October 1 the minimum number of refugees to be allowed into the US the following year. To support this, the [[w:Department of Health and Human Services|Department of Health and Human Services]] (HHS) produced a [[w:Cost–benefit analysis|cost-benefit analysis]] on the economic impact of refugees. A draft of this 2017 analysis concluded that “the net financial impact of refugees was positive over the 10-year period, at $63 billion.” Then [[w:Stephen Miller|Stephen Miller]] intervened to delete the benefits -- tax revenues -- generated by refugees. The resulting final report concluded that the &quot;per-capita refugee costs for major HHS programs [averaged over 10 years] totaled $3,300.” Miller’s [[w:United States Domestic Policy Council|Domestic Policy Council]] (DPC) reportedly &quot;maneuvered around the [[w:United States Department of State|State Department]], [[w:United States Department of Defense|Defense Department]], [[w:Joint Chiefs of Staff|Joint Chiefs of Staff]], [[w:Office of the Vice President of the United States|Office of the Vice-]]President, and [[w:Office of Management and Budget|Office of Management and Budget]] to push through his draconian cuts to the refugee cap, dismissing and rewriting the evidence as needed.&quot; A State Department official said, &quot;The process has never been this corrupt.” UCS reported that, &quot;Manipulating scientific evidence to fit a political narrative is a clear path toward implementing policies that are detrimental both to the US and to our allies.&quot;&lt;ref&gt;Union of Concerned Scientists (2018).&lt;/ref&gt;

To the extent that these comments provide a balanced summary of the research in this area, it suggests that leaders like Trump, Sessions, and Miller, to name only three, can be fairly describes as &quot;entrepreneurs of hate&quot;. 

We next discuss competition in media markets before discussing immigrants further. 

=== Diversity of ownership of the major media === 

Acemoglu and Johnson (2023, esp. ch. 4) describe how religious leaders in hierarchical societies prior to [[w:James VI and I|King James]] of the [[w:King James Version|King James Bible]] convinced commoners to live in poverty while giving increasing shares of what they produced, so religious and secular elites could live in opulence. Prior to King James the major media were religious organizations. In such hierarchical societies some religious leaders were entrepreneurs of hate, exhorting the faithful to fear and hate designated evil others and give increasing shares of what they produce so secular elites would &quot;protect&quot; them in wars against designated evil others.

Acemoglu and Robinson (2012, p. 461) said, &quot;Empowerment of society at large is difficult to coordinate and maintain without widespread information about whether there are economic and political abuses by those in power. ... Pamphlets and books informing and galvanizing people played an important role during the [[w:Glorious Revolution|Glorious Revolution]] [1688] in England, the French Revolution [1789-1799]&quot; and in the US. ([[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]] and [[w:James A. Robinson|Robinson]] shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]].)

But pamphlets started challenging established power structures much earlier. The publication of [[w:Martin Luther|Martin Luther]]'s ''[[w:Ninety-five Theses|Ninety-five Theses]]'' in Latin 1517-10-31 was quickly condemned by [[w:Pope Leo X|Pope Leo X]] and his hierarchy. [[w:Johann Tetzel|Johann Tetzel]], who had been selling [[w:Indulgence|indulgence]]s, published a pamphlet calling for Luther to be burnt for [[w:heresy|heresy]]. Eight hundred copies were reportedly sent to [[w:Wittenberg|Wittenberg]] to be sold, but students seized and burned them.&lt;ref&gt;Hendrix (2015, p. 64), quoted from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; This led to a pamphlet war. Luther's brief German-language reply was short and easy for commoners to understand&lt;ref&gt;Leppin and Wengert (2015, p. 389), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; and was much more popular than Tetzel's.&lt;ref&gt;Pettegree (2015, p. 145), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; 

Less than twenty years later, in 1534, King [[w:Henry VIII|Henry VIII]] declared himself head of the [[w:Church of England|Church of England]], denying the authority of the Pope. To secure the support of his subjects, he drove a propaganda war that included print literature&lt;ref&gt;Raymond (2006), cited from the Wikipedia article on &quot;[[w:Pamphlet war|pamphlet war]]&quot;, accessed 2026-10-15.&lt;/ref&gt; and theatre and minstrel troupes funded by the king. The latter traveled around the land to promote the new religious practices. Henry was portrayed as the glorious king of England, bravely and heroically defending the true faith against foreign devils, especially the Pope and Catholic priests, monks, etc.&lt;ref&gt;Hibbert et al. (2010, p. 928), cited from the Wikipedia article on &quot;[[w:Henry VIII|Henry VIII]]&quot;, accessed 2026-10-15.&lt;/ref&gt;

Henry died in 1547. Less than a dozen years later, the crown passed to [[w:Elizabeth I|Elizabeth I]]. Her limited personal wealth and English tradition required her to negotiate with Parliament for tax money. Acemoglu and Robinson (2012, pp. 105-111) noted that she was forced to limit the granting of monopolies and allow a class of merchants to compete in international trade. Many English merchants became wealthy, helped by the defeat of the [[w:Spanish Armada|Spanish Armada]] in 1588. 

When Elizabeth died in 1603, she was replaced by her cousin, King [[w:James VI and I|James VI of Scotland]], who then became also James I of England. He claimed [[w:Divine right of kings|divine right of kings]], threatening this class of merchants. To raise money without asking Parliament, he granted patents and monopolies to those who would pay. Parliament responded by passing the [[w:Statute of Monopolies|Statute of Monopolies]] on 1624-05-29. This statute repealed some past and future patents and monopolies except for novel inventions. 

But that Statute of Monopolies did not end the conflict. It ultimately went public, described as, &quot;A Pamphlet War in England, 1641-1643&quot;.&lt;ref&gt;Washington (2018).&lt;/ref&gt; Some pamphlets supported [[w:Charles I of England|Charles I]], who became King after James died in 1625. Others supported Parliamentarians and [[w:Levellers|Levellers]]. The latter supported radical ideas like [[w:popular sovereignty|popular sovereignty]], extended suffrage, equality before the law and religious tolerance. In 1642 war broke out between Parliament and Charles I, though many in Parliament sided with Charles -- especially those, who had received monopolies from him.&lt;ref&gt;Acemoglu and Robinson (2012, esp. pp. 189, 210).&lt;/ref&gt; The war was won by the [[w:New Model Army|New Model Army]], which represented many in Parliament plus Levellers. The latter were excluded from the government that followed the [[w:Execution of Charles I of England|beheading of King Charles I]] in 1649.&lt;ref&gt;Foxley (2013, p. 207), cited from the Wikipedia article on [[w:Levellers|Levellers]], accessed 2026-10-15.&lt;/ref&gt; However, that new government stopped granting as many monopolies and allowed so many commoners to become entrepreneurs that the GDP per capita began to grow at rates that are key features of Figure 1 for England and the 13 English colonies that became the US over a century later. 

One explanation for that economic growth was the diversity of independent publishers of pamphlets, books and newspapers. [[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]] of 1792&lt;ref&gt;Wikipedia &quot;[[w:Postal Service Act|Postal Service Act]]&quot;, accessed 2023-07-11.&lt;/ref&gt; made a major contribution to making the US what it is today.  Under that act, newspapers were delivered up to 100 miles for a penny, when postage for letters was between 6 and 25 cents depending on distance.&lt;ref&gt;Historian, USPS (2010).&lt;/ref&gt; McChesney and Nichols estimated that between 1840 and 1844, the US postal subsidy was a little over 0.2% of GDP.&lt;ref name=McC-N2010&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; 

During the first half of the nineteenth century the US had more independent newspaper publishers (more independent media organizations) per million population than at any other time or place in human history.&lt;ref&gt;John (1995). See also the interview with him at &quot;[[Media concentration per Columbia History Professor Richard John]]&quot; on Wikiversity, accessed 2026-10-12.&lt;/ref&gt; In 1932 with official unemployment over 20 percent, the conservative argument that blames the poor for their poverty did not sell newspapers, and the US still had sufficient competition in media markets that presidential candidate Franklin Roosevelt (FDR) got a mandate to spend the money to put humans back to work while taxing the ultra-wealthy like they had never been taxed before or since and during World War II to institute wage and price controls under the [[w:Office of Price Administration|Office of Price Administration]]. The latter prevented price gouging by government contractors that had generated inflation and stifled economic growth in previous major wars (World War I, the Civil War, and the War of 1812). Instead, the US averaged 7 percent per year growth for FDR's 12 years.&lt;ref&gt;The &quot;6 percent per year&quot; growth during FDR's 12 years was computed as the average of log(GDPpc) in the Maddison Data. To be precise, that computation produced 0.0597; converting that from a log scale to percent increase gives 0.0615. Using data from [[w:MeasuringWorth|MeasuringWorth]] instead of Maddison, we get 0.783 and 0.815, respectively. More documentation of this is available in &quot;[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]&quot; on Wikiversity, accessed 2026-10-13.&lt;/ref&gt; 

One of the most interesting features of Figure 1 is the behavior of the US economy in the administrations of [[w:Herbert Hoover|Herbert Hoover]] and FDR: Under Hoover, the economy declined at 9 percent per year for 4 years, a financial disaster that created the public demand for change that gave FDR the mandate to do what he did. Economic growth under two other presidents are numerically comparable to FDR's numbers but for much shorter periods: (1) [[w: Warren G. Harding|Warren G. Harding]] averaged 7 percent annual growth for two years before he died in 1923 during the [[w:Roaring Twenties|Roaring Twenties]]. (2) [[w:Rutherford B Hayes|Rutherford B Hayes]], who took office in 1877 at the end of the [[w:Panic of 1873|Panic of 1873]], oversaw 5.5 percent growth per year for his 4 years as president. Those impressive sounding average growth figures are so short that they are not noticeably different from other minor features visible in Figure 1. 

FDR also dramatically reduced inequality. For over a century between about 1820 and 1932 the top 1 percent of the US population owned roughly 45 percent of net personal wealth in the US, while the bottom 90 percent owned roughly 22 percent. Shortly after FDR died, the wealth of the top 1 percent and bottom 90 percent were even at roughly 30 percent. 

FDR also oversaw dramatic growth in the power of organized labor. Their successes continued to reduce inequality to roughly 22 percent for the top 1 percent vs. 37 percent for the bottom 90 percent when Ronald Reagan became president. Since then, [[w:Reaganomics|Reaganomics]] have been rebuilding the [[w:Robber baron (industrialist)|Robber Baron]]s.&lt;ref&gt;This history of inequality is documented in [[:File:Share of US wealth 90p99.svg|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023]] in the Wikiversity article on &quot;[[w:Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]&quot; in Wikiversity, accessed 2026-10-13.&lt;/ref&gt;

FDR made enemies among the wealthiest and most powerful. In a [[q:Franklin D. Roosevelt#Address at Madison Square Garden (1936)|speech in Maddison Square Garden in 1936]], FDR mentioned, &quot;the old enemies of peace: business and financial monopoly, speculation, reckless banking, class antagonism, sectionalism, war profiteering. ... They are unanimous in their hate for me -- and I welcome their hatred.&quot; They doubtless could see that they could no longer afford as many personal servants -- maids, butlers, cooks, chauffeurs, etc. -- as they had before FDR. 

Broadcasting was relatively new when FDR took office. The economics of broadcasting naturally favored a very few networks, whose power concerned FDR. In 1941 his [[w:Federal Communications Commission| Federal Communications Commission]] (FCC) adopted the [[w:Mayflower doctrine|Mayflower doctrine]], which prohibited broadcasters from advocacy. When FDR died, people with power were able to get the Mayflower doctrine replaced by the more lenient [[w:Fairness Doctrine|Fairness Doctrine]], which opened the floodgates for entrepreneurs of hate, who drove the [[w:McCarthyism|McCarthy Red Scare]] in the US and many other problems since. Over time this became worse, facilitated by increasing concentration of ownership of the media. That contributed to the end of the Fairness Doctrine starting in 1985. That revocation and especially the more recent rise of for-profit social media have amplified the ability of entrepreneurs of hate to make money both for commercial social media and for themselves by disseminating lies about others, e.g., immigrants.&lt;ref&gt;On how for-profit social media companies promote entrepreneurs of hate, see, e.g., DiResta (2026) and the Wikiversity article on, &quot;[[Managing the good and the evils of social media]]&quot;, accessed 2026-10-12. DiResta (2026, p. 1) described the promotion of &quot;influencers&quot; by for-profit social media as essentially unleashed &quot;a war for power, influence, and reality itself—a conflict in which the ends justify any means, and one that has ripped America apart, threatened our very democracy, and consumed my own life for the past decade, leading to death threats, congressional subpoenas, lawsuits, and legal bills running into seven figures.&quot; In the interview on Wikiversity, DiResta said her legal bills to that point had exceeded $7 million.&lt;/ref&gt; 

And the rate of economic growth in the US (and the Netherlands and the UK) have been slowing in recent decades, as is visible in Figure 1. Inequality expert [[w:Thomas Piketty|Piketty]] (2022, p. 139) noted, {{quote| 

In the United States, the national income per inhabitant rose at a rate ... of 2.2% between 1950 to 1990 when the top tax rate reached on average 72%.  The top rate was then cut in half, with the announced objective of boosting growth.  But in fact, growth fell by half, to 1.1% per annum between 1990 and 2020 ... . [A]ll the data at our disposal today suggest that virtually confiscatory tax rates have been an immense historical success. ...[T]he battle for equality and education ... has made economic development and human progress possible, and not the veneration of property, stability, and inequality.}}

Table 1 compares Piketty's growth numbers with computations based on the data from both Maddison and [[w:MeasuringWorth|MeasuringWorth]]: The numbers are close but not exact. 

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;&quot;
|+ Table 1. US growth rate 1950 to 1990 to 2020
|-
! 
! colspan=3 | US GDP per capita !! colspan=2 | Growth (% / yr)
|- 
! source || 1950 !! 1990 !! 2020 !! 1950 - 1990 !! 1990 - 2020 
|-
| Piketty || - || - || - || 2.2% || 1.1%  
|- 
| Maddison || $15,240 || $36,982 || $54,379 || 2.24% || 1.29%
|- 
| MeasuringWorth || $16,208 || $40,191 || $61,127 || 2.30% || 1.41%
|}

Acemoglu and Robinson (2012, pp.  105-106) provide a historical analysis supporting Piketty's concern about inequality slowing growth. They note that in 1588, England, Spain and France {{quote|

were ruled by relatively absolutist monarchs: [[w:Elizabeth I|Elizabeth I]] in England, [[w:Philip II of Spain|Philip II]] in Spain, and [[w:Henry III of France|Henry III]] in France. All were battling with assemblies of citizens -- such as the [[w:Parliament of England#Tudor era (1485–1603)|Parliament]] in England, the [[w:Cortes Generales#Habsburg rule (16th–17th centuries)|Cortes]] in Spain, and the [[w:Estates General (France)#Revival in 1560–1614|Estates-General]] in France -- that were demanding more rights and control over the monarchy. ... [T]he English Parliament and the Spanish Cortes had power over taxation, while the Estates-General did not. In Spain this mattered little, because after 1492 the Spanish Crown had a vast American empire and benefited massively from the gold and silver found there. In England ... Elizabeth I was far less financially independent, so she had to beg Parliament for more taxes. In exchange, Parliament demanded concessions, in particular restrictions of the right of Elizabeth to create monopolies. ... In Spain the Cortes lost a similar conflict.}} 

Consistent with Acemoglu and Johnson (2023) and the discussion above, inequality tends to increase elite control of the media, which it makes it harder for commoners to get a fair hearing of their concerns. 

=== Media conflicts of interest in IP law ===

Major media have [[w:Conflict of interest|conflicts of interest]] in honestly reporting on anything involving [[w:Intellectual property|intellectual property]] (IP) law. An example of Cardwell's law, mentioned above, is the [[w:Copyright Term Extension Act|Sonny Bono Copyright Term Extension Act]], also known as the Mickey Mouse Protection Act of 1998. The copyright term on ''[[w:Steamboat Willie|Steamboat Willie]]'', the first [[w:Mickey Mouse|Mickey Mouse]] animated film that Disney distributed, was extended multiple times by the US Congress following extensive lobbying by the film industry led by [[w:The Walt Disney Company|the Walt Disney Company]]. ''Steamboat Willie'', which debuted in 1928, finally entered the [[w:Public domain in the United States|US public domain]] on January 1, 2024. 

Eric Eldred and others filed suit, claiming that allowing any number of retroactive extensions violated the [[w:Copyright Clause|Copyright Clause]] of the US Constitution, which says, {{quote|

[the United States Congress shall have power] To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.}}

In ''[[w:Eldred v. Ashcroft|Eldred v. Ashcroft]]'' (2003), the [[w:Supreme Court of the United States|US Supreme Court]]  disagreed with Eldred and allowed the statute to remain in effect. However, that case drew enough public attention that copyright terms have not (yet) been extended further, and ''Steamboat Willie'' is currently in the pubic domain. 

[[w:Lawrence Lessig|Lessig]] (2004) ''[[w:Free Culture (book)|Free Culture]]'' describes other examples, where US copyright law seems to ''limit'' rather than &quot;promote the Progress of Science and useful Arts&quot;. This includes a case where a 2002 freshman, Jesse Jordan, modified a search engine to make it easier for other student to find pictures, research, notes, movie clips, etc., on an [[w:intranet|intranet]] at [[w:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] (RPI). The [[w:Recording Industry Association of America|Recording Industry Association of America]] (RIAA) sued Jordan and three other students for close to $100 billion for copyright violations -- six times the total profit of the film industry in 2001. Jordan had an uncle, who was an attorney, who explained that the legal bills for fighting that case would be at least $250,000. That would bankrupt his parents; they could not recover that money, even if they won. Jordan had saved $12,000 to cover part of his college expenses. RIAA accepted that and settled the case.&lt;ref&gt;Lessig (2004, p. 42).&lt;/ref&gt; 

This is an example of a [[w:Strategic lawsuit against public participation|strategic lawsuit against public participation]] (SLAPP), which in this case rested on questionable interpretations of copyright law. The major media have conflicts of interest in honestly reporting on anything involving (a) copyright laws, because much of their income is derived from copyright royalties, or (b) anything that might offend their major accounts, including major corporations. And anti-SLAPP laws, which exist in some jurisdictions but not others,&lt;ref&gt;[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel for the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]], supports anti-SLAPP laws and the &quot;JAWBONE act&quot;, to limit the power of the government to threaten questionable actions, like the Trump administration's threats against the ABC network over a Jimmy Kimmel monolog. See the Wikiversity article on &quot;[[Foundation for Individual Rights and Expression]]&quot;, accessed 2026-10-16.&lt;/ref&gt; limit the ability of major corporations to abuse their power against potential challengers without such deep pockets.

Similarly Graves (2023) asked, &quot;Do copyrights and paywalls on refereed academic journals violate the US constitution?&quot; No one expects to make money from copyright royalties for drafts submitted to refereed academic journals. They do it out of a felt need (a) to contribute to the future of humanity or (b) to build their reputation or (c) because promotions at many universities are based partly on publication records. Graves is not alone in his concern about this: Many European countries do not allow reports of research they fund to be published behind paywalls.&lt;ref&gt;Dimi Zagorski is concerned that copyright exceptions for scientific research are different in different European countries, and that's a problem for the Wikimedia Foundation. See the discussion of this in the Wikiversity article on &quot;[[Wikimedia concerns with European copyright rules including AI and scientific research]]&quot;. That's the opposite of US law, which grants patents to universities, research institutions, private companies, and individuals for innovations developed under government contracts, per the Bayh-Dole Act of 1980, described by Baker (2016, p. 80).&lt;/ref&gt;

Regarding patent law, the major media have conflicts of interest in discussing proposed changes affecting major accounts, but they make little use of patent law themselves. This seems to have contributed to the claim that, &quot;[O]ver the last four decades ... stronger patent protection seems to be associated with slower growth&quot; in the US, according to [[w:Dean Baker|Baker]] (2016, pp. 77-78). A potentially similar phenomenon appears in a comparison of 21 different European countries in an analysis that displays &quot;a positive relationship between anticorruption measures in a country and rate of innovation in that country as measured by patents per capita&quot;, discussed by Aghion et al. (2021, Figure 15.1b, pp. 292-293).&lt;ref&gt;Usher and Kim-Leffingwell (2022) documented how each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN), relatively small nonprofits independent of major media organizations and not funded by advertising, in a US federal court district was associated with on average 1.4 additional prosecutions for political corruption per year between 2003 and 2019. They found on average 10 federal prosecutions for political corruption for each federal court district per year. Thus, 1.4 additional prosecutions per year per district represents an increase of 14 percent per member of INN.&lt;/ref&gt;

=== Immigrants === 

We now return to discussing the initiation of economic growth following the beheading of Charles I in 1649 visible in Figure 1. Charles I was replaced by [[w:English Council of State|Lord Protector Cromwell]], who refused to authorize as many monopolies as his predecessors, as mentioned above. He also encouraged Jews and dissenting Christians to settle in England, as mentioned by Mokyr (2017, p. 53), cited above. 

But Mokyr is not alone in claiming that immigrants contribute to economic growth. Aghion et al. (2021, p. 266) cite Arkolakis et al. (2020) for explaining how the period 1880–1920 is noteworthy for two parallel evolutions. First, the United States caught up with England and France to become the most technologically advanced country in the world as well as the wealthiest in terms of GDP per capita. Second, the United States welcomed large numbers of immigrants, most notably from Europe. The purpose of the cited study was to demonstrate a causal connection between these two phenomena: in other words, to show that European immigration was responsible for the United States’ taking the lead in technology. ... More precisely, if the United States had completely closed its borders to all foreign immigrants in 1880, its per capita GDP would have been 30 percent lower than the level actually attained in 1920&quot;. 

=== Industrial Revolution ===

There seems to be a broad consensus that the [[w:Industrial Revolution|Industrial Revolution]] began in England. However, there is less agreement on when it began. 

Mokyr (2017, p.  3) says, &quot;The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society.&quot; He described &quot;three cultural elements&quot; required for &quot;sustainable economic growth propelled by technological progress&quot;: (1, 2) The belief that it was both ''possible'' and ''desirable''. (3) Supportive policy measures had to be implemented. &quot;[B]efore 1750 they did not produce anything like an Industrial Revolution or sustainable economic growth propelled by technological progress.&lt;ref&gt;Mokyr (2017, pp. 19-20).&lt;/ref&gt; Acemoglu (2026, p. 53) similarly mentioned the eighteenth century in saying, &quot;The breakthroughs in production technology that started in the middle of the eighteenth century in Britain unleashed the Industrial Revolution.&quot; 

Aghion et al. (2021, esp. pp. 29-30) offer a ''later'' date for the start of the Industrial Revolution: They report that, &quot;European economies prior to 1820 experienced a series of phases of growth and decline. This raises a question: How did the pre-1820 period differ from the takeoff that started with the first Industrial Revolution?&quot; Aghion et al. (2021, p. 20) describe the &quot;first Industrial Revolution&quot; as having been triggered by &quot;the invention of the steam engine by James Watt in the 1770s ... . The invention of electricity set off the second Industrial Revolution, whose golden age was in the first half of the twentieth century. This second revolution originated with the invention of the [[w:Incandescent light bulb|light bulb]] by [[w:Thomas Edison|Thomas Edison]] in 1879 and of the [[w:dynamo|dynamo]] by [[w:Werner von Siemens|Werner von Siemens]] in 1866.&quot;&lt;ref&gt;The dynamo was independently invented by [[w:Henry Wilde (engineer)|Henry Wilde]], [[w:Charles Wheatstone|Charles Wheatstone]], [[w:Samuel Alfred Varley|Samuel Alfred Varley]] and Siemens, all about the same time, according to the Wikipedia article on [[w:Dynamo#Practical designs|dynamo]], accessed 2026-10-18.&lt;/ref&gt;

The image in Figure 1 raises questions about these claims by Mokyr, Acemogu, and Aghion et al.: It shows growth followed by stagnation in Holland but not &quot;phases of growth and decline&quot; with the exception of wars and [[w:Herbert Hoover|Herbert Hoover]]. It also does ''not'' show the slope (i.e., rate of growth) increasing starting around 1750 or the 1770s or 1879 or 1886. Instead it shows sustained growth in GDP per capita for England / Great Britain / the UK starting ''abruptly'' after 1649 with economic disruptions during the reign of [[w:Anne, Queen of Great Britain|Queen Anne]] (1702-1714) and the [[w:War of the Spanish Succession|War of the Spanish Succession]] (1701-1714). This was followed by relatively steady but slower growth as England was ruled by Kings [[w:House of Hanover#Monarchs of Great Britain, Ireland, and Hanover|George I, II, and III]] (1714-1820). Another dramatic increase in the slope started abruptly about 1819. 

Acemoglu and Johnson (2023, chapter 1) are more consistent with Figure 1. They said that, &quot;the Industrial Revolution ... [was] rooted in institutional changes that started empowering the middling sort of English people from the sixteenth and seventeenth centuries onward.&quot; ... However, &quot;the quality of life of a European peasant circa 1700 was not much different from that of an Egyptian peasant two thousand or seven thousand years previously.&quot; (chapter 5) They also claim that, &quot;English national production per capita doubled from 1500 to 1700&quot;, (chapter 5), and &quot;Growth emerged slowly in the second half of the eighteenth century as the Industrial Revolution, based on major technological breakthroughs and their application in industry, took root.&quot; (p. 45, chapter 2)

Indeed, Figure 1 shows English GDP per capita doubling between 1500 and 1700, but it appears to have been fairly constant from 1500 to 1649, then roughly doubled in the next half century. [The data on England up to 1700 in Figure 1 came from Broadberry et al. (2015), which was not available when Acemoglu and Robinson were writing ''Why Nations Fail'', which appeared in 2012.] 

Acemoglu and Robinson (2012, ch. 11) describe the financial instability and slow growth during the reigns of Queen Anne and George I, II, and III. After the Glorious Revolution, France, Spain, the pope, and others refused to recognize [[w:William III of England|William and Mary]] (1689-1702) and then Queen Anne as heads of state of England. Instead they continued to support the claims of James II and then his son, [[w:James Francis Edward Stuart|James Francis Edward Stuart]] (the &quot;Old Pretender&quot;, a brother of Queen Anne), after James II died in 1701. During the War of the Spanish Succession, each side tried to exploit the internal divisions of the other. Owen (1938, ch. 8) said France almost succeeded in an attempt in 1708 to [[w:Planned French invasion of Britain (1708)|land between 5 and 6 thousand soldiers in northeast Scotland]]. [[w:Louis XIV|Louis XIV]] had hoped to get the Old Protector named as King James VIII of Scotland. There were still [[w:Jacobitism|Jacobites]] in Scotland, who supported the Old Pretender and likely would have supported the invaders if they had landed. If they had succeeded, it would have created major problems for Queen Anne and her supporters. However, the Royal Navy prevented the invaders from disembarking, and they returned to France. 

When Queen Anne died in 1702, she was succeeded by George I. His Parliament was dominated by the Whig party, which represented the new mercantile and economic interests and had been the main organization behind the Glorious Revolution. Once in power, they were tempted to use their newly found position to prey on the rights of others but were constrained both by competing groups in Parliament, &quot;and by the very institutions that they had fought to introduce to strengthen Parliament and to prevent the emergence of a new absolutism and the return of the Stuarts&quot;,&lt;ref&gt;After James II fled England, the new government worked to block [[w:Jacobitism|Jacobitism]], which attempted to restore the [[w:House of Stuart|House of Stuart]] to the thrones of Scotland and Britain. When James II died, his son became titular head of this movement, who became known as &quot;the [[w:James Francis Edward Stuart|Old Pretender]]&quot;. After he died, he was replaced by his son, who became known as &quot;the [[w:Charles Edward Stuart|Young Pretender]]&quot;.&lt;/ref&gt; according to Acemoglu and Robinson (2012, pp. 302-303). The Whigs gave &quot;[[w:Common land|common land]]&quot; to nobles, confiscating it from commoners. Some commoners, who had traditionally shared that land, joined &quot;Blacks&quot;, so called, because they blackened their faces when hunting on former commons or destroying the property of the nobility or extorting money from nobles or their employees. Parliament retaliated by passing the [[w:Black Act 1723|Black Act 1723]], under which someone accused of [[w:Poaching|poaching]] a [[w:Hare|hare]], for example, could under certain circumstances receive a summary judgment of guilty and then be executed without delay if apprehended later.&lt;ref&gt;Newman and Brown, eds. (1997, p. 58). See also the Wikipedia article on &quot;[[w:Black Act 1723|Black Act 1723]]&quot;, accessed 2026-10-17.&lt;/ref&gt; 

But not all accused were summarily executed. In 1724 &quot;John Huntridge was accused of aiding deer stealers and abetting known Blacks, both crimes punishable by hanging. The prosecution of Huntridge came right from the top, initiated by the Regency Council of Lords Justices ... . After a trial of eight or nine hours, the jury found Huntridge innocent, partly on procedural grounds, since there were irregularities with the way the evidence had been collected. ... The jury was made up not of Huntridge’s peers, but of major landowners and gentry, who ought to have sympathized&quot; with the noble, whose property had been attacked by Blacks allegedly aided by Huntridge.&lt;ref&gt;Acemoglu and Robinson (2012, pp. 304-305).&lt;/ref&gt; 

These kinds of actions by the Parliaments of George I, II, and III might explain the slower growth during that period compared to that of the last half of the seventeenth century. That slower growth ended abruptly at the time of the [[w:Peterloo massacre|Peterloo Massacre]], 1819-08-16, visible in Figure 1. The growth of factories implied growth of cities, where workers lived closer together and could more easily organize. A campaign arose with multiple demands: They opposed the [[w:Corn Laws|Corn Laws]], which kept prices high for cereal grains and products made from them. They also demanded &quot;universal [adult male] suffrage&quot; and secret ballots in elections. A series of public meetings discussing these grievances led to one 1819-08-16, which attracted roughly 60,000. Acemoglu and Robinson (2012, p. 207) reported that, {{quote|

The authorities were very nervous about the meeting, and a force of six hundred cavalry of the [[w:15th The King's Hussars|Fifteenth Hussars]] had been assembled. As the speeches began, a local magistrate decided to issue a warrant for the arrest of the speakers. As police tried to enforce the warrant, they met with the opposition of the crowd, and fighting broke out. At this point the Hussars charged the crowd. Within a few chaotic minutes, eleven people were dead and probably six hundred wounded. The ''[[w:Manchester Observer|Manchester Observer]]'' called it the Peterloo Massacre. But given the changes that had already taken place in economic and political institutions, long-run repression was not a solution in England. The Peterloo Massacre would remain an isolated incident. Following the riot, the political institutions in England gave way to the pressure, and the destabilizing threat of much wider social unrest, particularly after the 1830 revolution in France against [[w:Charles X of France|Charles X]], who had tried to restore the absolutism destroyed by the French Revolution of 1789. In 1832 the government passed the [[w:Reform Act 1832|First Reform Act]]. It enfranchised [[w:Birmingham|Birmingham]], [[w:Leeds|Leeds]], [[w:Manchester|Manchester]], and [[w:Sheffield|Sheffield]], and broadened the base of voting so that manufacturers could be represented in Parliament. ... [I]n 1846 they managed to get the hated Corn Laws repealed ... .}}

The ''Manchester Observer'' closed roughly two years later after [[w:Manchester Observer#Closure by repeated prosecution|repeated prosecutions]] for [[w:Seditious libel|seditious libel]]. However, the British government placed enough other limits on its power that enough other commoners were allowed to become successful entrepreneurs that the rate of economic growth in the UK increased immediately, as indicated by the increase in the slope of the UK line in Figure 1 until World Wars I and II. 

== In sum ==

The above suggests that the rate of economic growth changes with either (a) a change in administration -- a new head of state -- or (b) some other major event like war, the Peterloo Massacre, or the [[w:Wall Street crash of 1929|Wall Street crash of 1929]]. Policy changes that can increase growth include (1) limits on grants of monopolies, (2) diverse ownership and control of the major media and (3) support for immigrants. 

However, correlation does not imply causation. [[w:Randomized controlled trial|Randomized controlled trial]]s have made important contributions to human understanding of economic growth, as documented by Banerjee, Duflo, and Kramer, cited above and recommended in the Wikiversity article on &quot;[[Information is a public good: Designing experiments to improve government]]&quot;. Failing that, more analyses of data from different political jurisdictions with different media policies could be useful, as described by Neff and Pickard (2024).&lt;ref&gt;Graves (2025) includes an analysis of the data in Neff and Pickard (2024).&lt;/ref&gt; 

== Caveats ==

We should be more confident of the results of randomized controlled trials, as recommended by Banerjee, Duflo, and Kremer, than of anyone interpreting the historical record, even by Nobel economists. 

The above attempts to accurately reflect the perspectives of all sources cited. Readers are invited to revise this article, subject to the standard Wikimedia rules of writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”{{'}}''' page]] -- and treating others with respect. 

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!-- Daron Acemoglu (2026-08-11) What Happened to Liberal Democracy?: Remaking a Politics of Shared Prosperity (Function)--&gt;{{cite Q|Q141241291}}
* &lt;!--Daron Acemoglu and Simon Johnson (2023) Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity (PublicAffairs)--&gt;{{cite Q|Q125292212}}
* &lt;!--Daron Acemoglu and James A. Robinson (2012) Why Nations Fail (Crown)--&gt;{{cite Q|Q7997840}} 
* &lt;!--Philippe Aghion, Céline Antonin, and Simon Bunel (2021) The Power of Creative Destruction: Economic Upheaval and the Wealth of Nations (Harvard University Press)--&gt;{{cite Q|Q137641358}}
* &lt;!--Ajzenman, Nicolas (29 January 2023). &quot;Migrants don't cause crime rates to increase — but false perceptions endure anyway&quot;. The Conversation.--&gt;{{cite Q|Q141758664}}
* &lt;!--Samuel Alexander (2026-07) &quot;Game Theory and the Dilemma of Degrowth: Why Embrace Degrowth if Other States or Societies Do Not?&quot;--&gt;{{cite Q|Q141968019}}
* &lt;!--Costas Arkolakis, Michael Peters, and Sun Kyoung Lee, (2020) &quot;European Immigrants and the United States’ Rise to the Technological Frontier&quot;--&gt;{{cite Q|Q141678287}}
* &lt;!--Dean Baker (2016) Rigged: How globalization and the rules to the modern economy were structured to make the rich richer--&gt;{{cite Q|Q100216001}}
* &lt;!--Abhijit V. Banerjee and Esther Duflo (2019) Good Economics for Hard Times (PublicAffairs)--&gt;{{cite Q|Q85764011}}
* &lt;!--Jutta Bolt and Jan Luiten van Zanden (2024-09-18) &quot;Maddison style estimates of the evolution of the world economy: A new 2023 update&quot;, Journal of Economic Surveys, 1-41--&gt;{{cite Q|Q126723821}}
* &lt;!--S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)--&gt;{{cite Q|Q57945943}}
* &lt;!--Donald S. L. Cardwell (1972) Turning Points in Western Technology (Neale Watson, Science History Pubs)--&gt;{{cite Q|Q141795083}}
* &lt;!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition--&gt;{{cite Q|Q141156783}}
* &lt;!--Renée DiResta and Rachel Kleinfeld (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies--&gt;{{cite Q|Q141158362}}
* &lt;!--Katarzyna Dośpiał-Borysiak (2017) &quot;Model of state management of petroleum sector -- Case of Norway--&gt;{{cite Q|Q141774179}}
* &lt;!--Rachel Foxley (2013) The Levellers: Radical Political Thought in the English Revolution (Oxford University Press)--&gt;{{cite Q|Q136227988}}
* &lt;!--Benjamin Gonzalez O’Brien, Loren Collingwood, Stephen Omar El-Khatib (2017-05-07) &quot;The Politics of Refuge: Sanctuary Cities, Crime, and Undocumented Immigration--&gt;{{cite Q|Q55932800}}
* &lt;!--Graves (2025-01-08) Defend Free Speech For All (Town Hall)--&gt;{{cite Q|Q132855526}}
* &lt;!--Graves (2023-03-21)  Do copyrights and paywalls on academic journals violate the US Constitution?--&gt;{{cite Q|Q126116187}}
* &lt;!--Christopher Hibbert, Ben Weinreb, John Keay, Julia Keay, and Matthew Weinreb (2011-09-09) The London Encyclopaedia, 3rd ed.--&gt;{{cite Q|Q137188993}}
* &lt;!-- Historian, USPS (2010) Postage Rates for Periodicals --&gt;{{cite Q|Q112167199}}
* &lt;!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse--&gt;{{cite Q|Q54641943}}
* &lt;!--Volker Leppin and Timothy J. Wengert (2015) &quot;Sources for and against the Posting of the Ninety-Five Theses&quot;--&gt;{{cite Q|Q141737347}}
* &lt;!--Lawrence Lessig (2004) Free Culture--&gt;{{cite Q|Q830526}}
* &lt;!--Olivier Marie and Paolo Pinotti (2024-02-01) &quot;Immigration and Crime: An International Perspective&quot;, Journal of Economic Perspectives. 38 (1): 181–200--&gt;{{cite Q|Q128434874}}
* &lt;!--Martínez, Daniel E.; Martínez-Schuldt, Ricardo D.; Cantor, Guillermo (2017). &quot;
Providing Sanctuary or Fostering Crime? A Review of the Research on 'Sanctuary Cities' and Crime&quot;
. Sociology Compass. 12 e12547. doi:10.1111/soc4.12547. ISSN 1751-9020.--&gt;{{cite Q|Q55932802}}
* &lt;!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) --&gt;{{cite Q|Q104888067}}.
* &lt;!-- Robert W. McChesney; John Nichols (2016). People get ready--&gt;{{cite Q|Q87619174}}
* &lt;!--Robert Michels (1911)  Les Partis politiques, essai sur les tendances oligarchiques des démocraties--&gt;{{cite Q|Q7210216}}
* &lt;!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Princeton University Press)--&gt;{{cite Q|Q136118590}}
* &lt;!--Joel Mokyr (1994) &quot;Cardwell's Law and the political economy of technological progress&quot;, Research Policy 23(5)561-574--&gt;{{cite Q|Q139772607}}
* &lt;!--Ashley N. Muchow &amp; Taisiia Stanishevska (2026) &quot;From Headlines to Handcuffs: Local News Coverage of Immigrant Crime and Hispanic Arrest&quot;, Justice Quarterly, vol. 43(3) (https://doi.org/10.1080/07418825.2025.2550307)--&gt;{{cite Q|Q141760694}}
* &lt;!--Neff and Pickard (2024) &quot;Funding Democracy: Public Media and Democratic Health in 33 Countries&quot;--&gt;{{cite Q|Q131468289}}
* &lt;!--Gerald G. Newman and Leslie Ellen Brown, eds. (1997) Britain in the Hanoverian Age, 1714–1837: An Encyclopedia--&gt;{{cite Q|Q141759196|authors = Gerald G. Newman and Leslie Ellen Brown, eds.}}
* &lt;!--John Hely Owen (1938)  War at Sea Under Queen Anne 1702–1708--&gt;{{cite Q|Q141757101}}
* &lt;!--John Perkins (2004) Confessions of an Economic Hit Man--&gt;{{cite Q|Q400098}}
* &lt;!--Andrew Pettegree (2015-10-27) Brand Luther: How an Unheralded Monk Turned His Small Town into a Center of Publishing, Made Himself the Most Famous Man in Europe--and Started the Protestant Reformation--&gt;{{cite Q|Q141761223}}
* &lt;!--Thomas Piketty (2022) A Brief History of Equality--&gt;{{cite Q|Q139886018}}
* &lt;!--Joad Raymond (2006-11-02) Pamphlets and Pamphleteering in Early Modern Britain--&gt;{{cite Q|Q141761849}}
* &lt;!--James A. Robinson (2025-04) &quot;Paths to the periphery&quot;, NBER Working Paper 33671--&gt;{{cite Q|Q141967690}}
* &lt;!--Nick Roll (2017-07-16) &quot;Correcting Jeff Sessions: Attorney general railed against crime in sanctuary cities, citing two professors’ work. But his claims are contrary to their conclusions.--&gt;{{cite Q|Q141735231}}
* &lt;!--J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)--&gt;{{cite Q|Q137669960}}
* &lt;!--I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)--&gt;{{cite Q|Q137669987}}
* &lt;!--Kelly Swanson (2017-09-17) &quot;The Trump administration rejects its own study finding refugees help the economy--&gt;{{cite Q|Q141900700}}
* &lt;!--Ted Trainer (2025-03) &quot;The 2024 Nobel Prize for Economics: An example of how conventional economics misrepresents reality--&gt;{{cite Q|Q142065447}}
* &lt;!--Union of Concerned Scientists (2019-01-11) &quot;The Trump Administration Manipulates Study on Economic Benefits of Refugees in US&quot;--&gt;{{cite Q|Q141898260|author=Union of Concerned Scientists}}
* &lt;!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption --&gt;{{Cite Q|Q134715465}}
* &lt;!--J. L. van Zanden  and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30--&gt;{{cite Q|Q137670058}} 
* &lt;!--Brittney Washington (2018-06-19) &quot; A Pamphlet War in England, 1641-1643&quot;--&gt;{{cite Q|Q141822147}}
* &lt;!--Tom K. Wong (2017-01-26) &quot;The Effects of Sanctuary Policies on Crime and the Economy&quot;--&gt;{{cite Q|Q98775875}}

[[Category:Original research]]
[[Category:Research]]
[[Category:Economics]]
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[[Category:Gross domestic product]]
[[Category:Economic growth]]
[[Category:Modern Monetary Theory]]
[[Category:Political science]]
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:''This article invites all to contribute references and interpretations to the growth in average annual income, Gross Domestic Product (GDP) per capita at purchasing power parity, as applied to leading countries in the [[w:Maddison Project|Maddison Historical Statistics Project]].''
:''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”'''{{'}} page]] -- and treating others with respect.''  
::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]''

== Abstract ==

This article first identifies and plots GDP per capita for leading countries in the Maddison Project data then discusses comments by Nobel prize winning economists and others to interpret what appears in said plot. No one really knows how to make economies grow. However, growth at the technology frontier requires creative destruction, and immigrants tend to be more entrepreneurial than native born. Also, creative destruction is facilitated and inequality is limited by diversity of ownership and control of the major media. 

== Maddison Historical Statistics == 

[[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD = initially [[w:Holland|Holland]] to 1807 and the [[w:Netherlands|Netherlands]] after; GBR = initially England to 1700, Great Britain to 1870, then the UK; USA starting in 1650 with the 13 colonies that became the US; SGP = Singapore starting in 1870).&lt;ref&gt;See Bolt and van Zanden (2024) for the Maddison Data generally, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, Sugimoto (2011) for Singapore to 2007, van Zanden and van Leeuwen (2012) for the data on Holland 1349–1807, and Smits et al. (2000) for the data on the Netherlands 1808-1913.&lt;/ref&gt; The dramatic drop in the orange line between 1807 and 1808 is due to the change in reference from Holland to the Netherlands; at that time Holland was the most prosperous region of the Netherlands.&lt;ref name=vanZ_B&gt;Private email exchange with van Zanden and Bolt, 2026-01-08.&lt;/ref&gt;'']]

The [[w:Maddison Project|Maddison Historical Statistics Project]] (aka Maddison Project) is widely recognized as the premier data source on historical macroeconomic data. It is named after [[w:Angus Maddison|Angus Maddison]] (1926-2010), who was a pioneer in [[w:National accounts|national accounts]], especially estimating [[w:Gros Domestic Product|gross domestic product]] (GDP) per capita (GDPpc) and population (pop) at various times dating back to the year 1 of the [[w:Common Era|Common Era]] (CE / AD). After Maddison's death, the Maddison Project has continued to update the database to include the best sources available on any country for any historical period since year 1. 

When accessed 2026-10-10 the version of the data available had been released 2024-09-18 containing data to 2022.&lt;ref&gt;Bolt and van Zanden (2024).&lt;/ref&gt; This dataset includes 23,280 observations on at least one of GDPpc and pop for an average of 137 years for 169 countries. Only 18 countries had the highest GDPpc for at least one year. Annual data starts in 1252, and that's only for one country, GBR = England / Great Britain / the United Kingdom. NLD (Holland / the Netherlands) had the highest GDPpc for 97 percent of the years between 1349 and 1807. The drop in the orange line in Figure 1 between 1807 and 1808 is due to the change of reference from Holland to the Netherlands.&lt;ref name=vanZ_B/&gt; Visual comparison of the orange and red lines in Figure 1 suggests that the GDPpc of Great Britain overtook that of the Netherlands a few decades earlier and overtook Holland a few decades later. This analysis focuses on data since 1349, in part because data were available for at most 4 countries for most years prior to that date. (The 3-letter country codes used here are [[w:ISO 3166-1 alpha-3|ISO 3166-1 alpha-3]], used in the Maddison data.) 

We are interested in technology leaders, so we eliminate the [[w:petrostates|petrostates]] of Qatar (QAT), Kuwait (KWT), and United Arab Emirates (ARE), plus Norway (NOR); the latter owes a substantial portion of their wealth to democratic management of [[w:North Sea oil|North Sea oil]].&lt;ref&gt;Dośpiał-Borysiak (2017).&lt;/ref&gt; With these restrictions 12 countries had the highest GDPpc for at least one year. However, FRA (France), ITA (Italy), SWE (Sweden), BEL (Belgium), NZL (New Zealand) and CHE (Switzerland) led for at most 7 of the 674 years between 1349 and 2022; we eliminate those, because including them would seem to add more clutter than clarity. We also eliminate AUS (Australia), which led for 17 of the 39 years between 1853 and 1891, when it was competing for the leadership with GBR and USA, for the same reason. And we eliminate LUX (Luxembourg), which has a population under a million. 

This leaves NLD, GBR, USA, and SGP (Singapore), plotted in Figure 1; Singapore has led since 2010.&lt;ref&gt;More details of this analysis are available in the chapter on &quot;[[Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs|The impact of the media on political economy since the time of the Pharaohs]]&quot;&quot; in the book-in-progress on ''[[Media Literacy and You]]'' on Wikiversity.&lt;/ref&gt; This plot has some interesting features, which we discuss in the remainder of this research report. 

== Nobel economists and others ==

[[w:Abhijit Banerjee|Banerjee]] and [[w:Esther Duflo|Duflo]] (2019, p. 207) said, &quot;despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive.&quot; That appeared shortly before they shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]] with [[w:Michael Kremer|Michael Kremer]] for documenting how [[w:Randomized controlled trial|randomized controlled trial]]s could be used to alleviating poverty in [[w:Development economics|development economics]].&lt;ref&gt;For the present discussion, we assume that enough [[w:Externality|externalities]] have been sufficiently internalized with appropriate [[w:Pigouvian tax|Pigouvian tax]]es to make growth desirable. This converts discussions of [[w:Degrowth|degrowth]], e.g., by Alexander (2026), into evaluations of whether externalities have actually been sufficiently internalized and what to do if they are not. Internalizing externalities appropriately will not be easy as long as those who benefit from not paying for damages they cause to customers, employees and neighbors control substantial portions of the funds for the media. However, for the present discussion, we assume this is not an issue.&lt;/ref&gt; 

While Banerjee and Duflo insisted that no one knows how to make an economy grow, other leading economists have recommended partial solutions. For example, [[w:Joel Mokyr|Mokyr]] (2017, p. 53) said, &quot;The willingness of the Netherlands and later Britain and the United States to tolerate Jews and dissenting Christians contributed a great deal to their economies, especially in high-skilled manufacturing and financial services.&quot; In other words, immigrants made an outsized contribution to the growth of NLD, GBR, and US, visible in Figure 1. Mokyr's comment appeared over a year before Banerjee and Duflo (2019) and roughy eight years before Mokyr shared the 2025 Nobel Memorial Prize in Economics with [[w:Philippe Aghion|Aghion]] and [[w:Peter Howitt (economist)|Howitt]]. 

Mokyr (2017, p. 255) also said that, &quot;intellectual innovation will eventually end up being increasingly resisted by an incumbent technology; if that resistance is not overcome, technological progress may be extinguished.&quot; That resistance, he suggested, drives {{quote|

the empirical regularity that no society remains at the cutting edge of technological creativity for very long&lt;ref&gt;Mokyr (2017, p. 260).&lt;/ref&gt;.}}

Mokyr (2017, p. 260; 1994) called this &quot;Cardwell's Law&quot; after Cardwell (1972), which describes {{quote|

the interaction between technological creativity and political competition ... based on the observation that technological progress encounters resistance from various groups that believe they stand to lose from innovation. These pressure groups will try to manipulate the political system to suppress successful innovation.}}

[[w:Robert Michels|Michels]] (1911) called this &quot;the [[w:Iron law of oligarchy|iron law of oligarchy]]&quot;. Further documentation of this appears in Acemoglu and Robinson (2012) and Acemoglu and Johnson (2023), who shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]]. 

Figure 1 begins with growth in England and Holland between 1349 and 1400, likely due to the fact that the [[w:Black Death|Black Death]] (1346-1353) killed between 30 and 60 percent of the population: Since land was not destroyed, survivors were likely able to produce more agricultural products for the same labor. Also, under the feudal system in place before the Black Death, commoners were not allowed to move without the permission of their employer, the Lord of the land where they worked. Efforts by elites to prevent commoners from moving and obtaining higher wages failed in England.&lt;ref&gt;Acemoglu and Robinson (2012, esp. pp. 99-101) note that surviving commoners in England were able to bargain for higher wages. The authorities passed emergency legislation in 1349 and 1351 that made it a crime to offer or accept wages higher than before the plague or to refuse to work or break an existing contract. In 1361 the penalties were increased to include imprisonment and even branding. This led to [[w:Peasants' Revolt|a major insurrection in 1381]] led by [[w:Wat Tyler|Wat Tyler]], which was put down with over 1,300 deaths. After that insurrection, authorities in England allowed commoners to demand and accept higher wages. Commoners were also allowed to move without the permission of their Lord, which had ''not'' been accepted before. By contrast, the treatment of commoners in Russia was reportedly ''worse'' than before the plague. Changes in the treatment of commoners between England and Russia seemed to vary depending on proximity to these extremes, with Austria more like Russia and France more like England.&lt;/ref&gt; The growth in Holland from 1492 has been has been attributed to [[w:Expulsion of Jews from Spain|Jews expelled from Spain settling]] in Holland and starting new businesses. 

The iron law of oligarchy provides an explanation for why growth stopped in England around 1400, Holland around 1600, and why growth has been slowing in recent decades in all the countries plotted in in Figure 1. It can also explain [[w:History of the Roman Empire|the rise and fall of the Roman Empire]], of the [[w:Maya civilization|Maya civilization]], and other societies that became very rich from allowing commoners to become entrepreneurs and then declined after previous winners changed the rules of the political economy to block further [[w:creative destruction|creative destruction]]. This includes how [[w:History of the Republic of Venice|Venice]] became &quot;possibly the richest place in the world, with the most advanced set of inclusive economic institutions&quot;, dominating trade in the [[w:Mediterranean Sea|Mediterranean]] based on a system that encouraged entrepreneurial commoners to undertake risky trading ventures starting after 810 CE and ending when leaders made wealthy by such ventures changed the rules to end that system starting in 1297, destroying their dominance of Mediterranean trade &quot;and converting Venice into a museum&quot;, in the words of Acemoglu and Robinson (2012, pp. 152-155).

Acemoglu and Robinson (2012) suggest three primary distinctions between the poorest and wealthiest nations. The poorest tend to be [[w:failed state|failed states]] like [[w:Somalia|Somalia]], which are poor, because of a substantial lack of rule of law and continued violent conflict between warring groups. Acemoglu and Robinson divide other nation states between &quot;extractive&quot; and &quot;inclusive&quot; regimes. &quot;Extractive&quot; regimes tend to be poor, because elites write the rules to protect their interests against the competition that generates the creative destruction required for economic growth at the technology frontier. &quot;Inclusive&quot; regimes limit monopolies and encourage enough commoners to become entrepreneurs that over time the economy tends to produce more with less, increasing the wealth of the nation. 

[[w:Ted Trainer|Trainer]] (2025) described the 2024 Nobel Prize for Economics as, &quot;An example of how conventional economics misrepresents reality&quot;, blaming the poor for their poverty, e.g., in failing to cite [[w:John Perkins (author)|Perkins]] (2004) ''[[w:Confessions of an Economic Hit Man|Confessions of an Economic Hit Man]]''. Doth professor Trainer protest too much? Chapter 9 in Acemoglu and Robinson (2012) describes, &quot;Reversing Development: How European colonialism impoverished large parts of the world&quot;, which describes several historical cases where a colonial power subordinated a local extractive elite, increasing the severity of their extractive policies, further impoverishing the vast majority of that locality. The [[w:Dutch East India Company|Dutch East India Company]] did this to most of [[w:Indonesia|Indonesia]], but they could not do this in the [[w:Banda Islands|Banda Islands]], which lacked a central authority whom the Dutch could coerce into signing a monopoly treaty. Politics there were more inclusive, with many small autonomous city-states, run by village meetings of citizens. To avoid having to compete with English, Portuguese, Indian, and Chinese merchants, the Dutch massacred &quot;almost the entire population of the islands, probably about fifteen thousand people. ... [O]nly a few were left alive, enough to preserve the know-how necessary for mace and nutmeg production. After this genocide was complete,&quot; the Dutch repopulated the Banda islands with a hierarchical plantation society.&lt;ref&gt;Acemoglu and Robinson (2012, p. 248).&lt;/ref&gt; 

One of the few places in the colonized world that successfully avoided this fate is [[w:Botswana|Botswana]]. In 1895 chiefs of three of the eight [[w:Tswana people|Tswana states]] in what is now Boswana sailed to England with a charm offensive to save their country from [[w:Cecil Rhodes|Cecil Rhodes]]. They met with [[w:Joseph Chamberlain|Joseph Chamberlain]], then colonial secretary, before and after an extensive speaking tour of England. With the favorable coverage that the chiefs got in the English press, Rhodes got a railroad through their country and nothing more. When Botswana gained independence in 1966, they had relatively inclusive political institutions that allowed them to grow economically, escaping the fate of Black Africans in [[w:Rhodesia|Rhodesia]] / [[w:Zimbabwe|Zimbabwe]] to their north and [[w:South Africa|South Africa]] to their south.&lt;ref&gt;Acemoglu and Robinsion (2012, pp. 404-405).&lt;/ref&gt; 

Robinson (2025) provided a related explanation for the poverty of Zimbabwe, whose 2026 population is roughly 17 million with a 2025 GDP per capital of roughly $5,400 at PPP. '[D]uring fieldwork in Zimbabwe in 2017 I had breakfast with the CEO of a subsidiary of a large multinational company. I was trying to understand the political incentives and mechanisms that had led to extractive economic institutions in the country, leading it to be substantially poorer today than it was at the time of independence in 1980. ... At some point in the conversation the CEO looked up at me and said: “I really don’t know what your problem is, it really isn’t complicated, what is it that you can’t understand? ... [T]here’s plenty of money in this country for 300 people.”'

Mokyr (2017, p. 60) also mentioned &quot;entrepreneurs of hate&quot;, who convince others to dislike some group in a way that may benefit those &quot;entrepreneurs&quot; and major media organizations, who promote such &quot;entrepreneurs&quot; to please those who control much of the money for the media. For-profit social media have been particularly effective in herding their audiences into echo chambers, reinforcing their preconceptions in ways that increase political polarization and violence and threaten democracy.&lt;ref&gt;See, e.g., DiResta (2026), DiResta and Keinfeld (2025), and the Wikiversity article on &quot;[[Managing the good and the evils of social media]]&quot;, accessed 2026-10-14.&lt;/ref&gt; When successful this protects current elites from competition that would otherwise drive [[w:creative destruction|creative destruction]], benefitting the public at large while threatening existing elites.  

For example, most research on immigrants suggest that on average immigrants are more entrepreneurial and no more likely to commit crime than native born,&lt;ref&gt;e.g., Martínez et al. (2017); Muchow and Stanishevska (2026); Ajzenman(2023).&lt;/ref&gt; and some studies report that [[w:Sanctuary city|sanctuary jurisdiction]]s have on average higher median incomes and less crime.&lt;ref&gt;Martínez et al. (2017); Wong (2017). See also the Wikipedia article on &quot;[[w:Sanctuary city|sanctuary city]]&quot;, accessed 2026-10-12.&lt;/ref&gt; Where immigrant crime is higher, it typically result from anti-immigrant labor market restrictions or from young less-educated males with irregular status.&lt;ref&gt;Marie and Paolo (2024). See also the Wikipedia article on &quot;[[w:Immigration and crime|Immigration and crime]]&quot;, accessed 2026-10-12.&lt;/ref&gt; 

But major media outlets make money from featuring charismatic figures, who attract an audience without offending people who control much of the money for the media. In 2017 President [[w:Immigration policy of the first Trump administration|Trump]]'s Attorney General [[w:Jeff Sessions|Jeff Sessions]] quoted a research study that supposedly showed that sanctuary cities had more crime than cities without sanctuary policies. The authors of that study insisted that Sessions had ''misrepresented'' their work, which concluded &quot;that sanctuary policies have no effect on crime rates, despite narratives to the contrary.&quot;&lt;ref&gt;Roll (2017); Gonzalez O’Brien, et al. (2017).&lt;/ref&gt; Similarly, a 2017-09-19 report described how, &quot;The Trump administration rejects its own study finding refugees help the economy&quot;. &lt;ref&gt;Swanson (2017).&lt;/ref&gt; Just under three months later, the [[w:Union of Concerned Scientists|Union of Concerned Scientists]] (UCS) described how, &quot;The Trump Administration Manipulates Study on Economic Benefits of Refugees in US&quot;: The [[w:Refugee Act of 1980|Refugee Act of 1980]]  requires the president to set by October 1 the minimum number of refugees to be allowed into the US the following year. To support this, the [[w:Department of Health and Human Services|Department of Health and Human Services]] (HHS) produced a [[w:Cost–benefit analysis|cost-benefit analysis]] on the economic impact of refugees. A draft of this 2017 analysis concluded that “the net financial impact of refugees was positive over the 10-year period, at $63 billion.” Then [[w:Stephen Miller|Stephen Miller]] intervened to delete the benefits -- tax revenues -- generated by refugees. The resulting final report concluded that the &quot;per-capita refugee costs for major HHS programs [averaged over 10 years] totaled $3,300.” Miller’s [[w:United States Domestic Policy Council|Domestic Policy Council]] (DPC) reportedly &quot;maneuvered around the [[w:United States Department of State|State Department]], [[w:United States Department of Defense|Defense Department]], [[w:Joint Chiefs of Staff|Joint Chiefs of Staff]], [[w:Office of the Vice President of the United States|Office of the Vice-]]President, and [[w:Office of Management and Budget|Office of Management and Budget]] to push through his draconian cuts to the refugee cap, dismissing and rewriting the evidence as needed.&quot; A State Department official said, &quot;The process has never been this corrupt.” UCS reported that, &quot;Manipulating scientific evidence to fit a political narrative is a clear path toward implementing policies that are detrimental both to the US and to our allies.&quot;&lt;ref&gt;Union of Concerned Scientists (2018).&lt;/ref&gt;

To the extent that these comments provide a balanced summary of the research in this area, it suggests that leaders like Trump, Sessions, and Miller, to name only three, can be fairly describes as &quot;entrepreneurs of hate&quot;. 

We next discuss competition in media markets before discussing immigrants further. 

=== Diversity of ownership of the major media === 

Acemoglu and Johnson (2023, esp. ch. 4) describe how religious leaders in hierarchical societies prior to [[w:James VI and I|King James]] of the [[w:King James Version|King James Bible]] convinced commoners to live in poverty while giving increasing shares of what they produced, so religious and secular elites could live in opulence. Prior to King James the major media were religious organizations. In such hierarchical societies some religious leaders were entrepreneurs of hate, exhorting the faithful to fear and hate designated evil others and give increasing shares of what they produce so secular elites would &quot;protect&quot; them in wars against designated evil others.

Acemoglu and Robinson (2012, p. 461) said, &quot;Empowerment of society at large is difficult to coordinate and maintain without widespread information about whether there are economic and political abuses by those in power. ... Pamphlets and books informing and galvanizing people played an important role during the [[w:Glorious Revolution|Glorious Revolution]] [1688] in England, the French Revolution [1789-1799]&quot; and in the US. ([[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]] and [[w:James A. Robinson|Robinson]] shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]].)

But pamphlets started challenging established power structures much earlier. The publication of [[w:Martin Luther|Martin Luther]]'s ''[[w:Ninety-five Theses|Ninety-five Theses]]'' in Latin 1517-10-31 was quickly condemned by [[w:Pope Leo X|Pope Leo X]] and his hierarchy. [[w:Johann Tetzel|Johann Tetzel]], who had been selling [[w:Indulgence|indulgence]]s, published a pamphlet calling for Luther to be burnt for [[w:heresy|heresy]]. Eight hundred copies were reportedly sent to [[w:Wittenberg|Wittenberg]] to be sold, but students seized and burned them.&lt;ref&gt;Hendrix (2015, p. 64), quoted from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; This led to a pamphlet war. Luther's brief German-language reply was short and easy for commoners to understand&lt;ref&gt;Leppin and Wengert (2015, p. 389), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; and was much more popular than Tetzel's.&lt;ref&gt;Pettegree (2015, p. 145), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.&lt;/ref&gt; 

Less than twenty years later, in 1534, King [[w:Henry VIII|Henry VIII]] declared himself head of the [[w:Church of England|Church of England]], denying the authority of the Pope. To secure the support of his subjects, he drove a propaganda war that included print literature&lt;ref&gt;Raymond (2006), cited from the Wikipedia article on &quot;[[w:Pamphlet war|pamphlet war]]&quot;, accessed 2026-10-15.&lt;/ref&gt; and theatre and minstrel troupes funded by the king. The latter traveled around the land to promote the new religious practices. Henry was portrayed as the glorious king of England, bravely and heroically defending the true faith against foreign devils, especially the Pope and Catholic priests, monks, etc.&lt;ref&gt;Hibbert et al. (2010, p. 928), cited from the Wikipedia article on &quot;[[w:Henry VIII|Henry VIII]]&quot;, accessed 2026-10-15.&lt;/ref&gt;

Henry died in 1547. Less than a dozen years later, the crown passed to [[w:Elizabeth I|Elizabeth I]]. Her limited personal wealth and English tradition required her to negotiate with Parliament for tax money. Acemoglu and Robinson (2012, pp. 105-111) noted that she was forced to limit the granting of monopolies and allow a class of merchants to compete in international trade. Many English merchants became wealthy, helped by the defeat of the [[w:Spanish Armada|Spanish Armada]] in 1588. 

When Elizabeth died in 1603, she was replaced by her cousin, King [[w:James VI and I|James VI of Scotland]], who then became also James I of England. He claimed [[w:Divine right of kings|divine right of kings]], threatening this class of merchants. To raise money without asking Parliament, he granted patents and monopolies to those who would pay. Parliament responded by passing the [[w:Statute of Monopolies|Statute of Monopolies]] on 1624-05-29. This statute repealed some past and future patents and monopolies except for novel inventions. 

But that Statute of Monopolies did not end the conflict. It ultimately went public, described as, &quot;A Pamphlet War in England, 1641-1643&quot;.&lt;ref&gt;Washington (2018).&lt;/ref&gt; Some pamphlets supported [[w:Charles I of England|Charles I]], who became King after James died in 1625. Others supported Parliamentarians and [[w:Levellers|Levellers]]. The latter supported radical ideas like [[w:popular sovereignty|popular sovereignty]], extended suffrage, equality before the law and religious tolerance. In 1642 war broke out between Parliament and Charles I, though many in Parliament sided with Charles -- especially those, who had received monopolies from him.&lt;ref&gt;Acemoglu and Robinson (2012, esp. pp. 189, 210).&lt;/ref&gt; The war was won by the [[w:New Model Army|New Model Army]], which represented many in Parliament plus Levellers. The latter were excluded from the government that followed the [[w:Execution of Charles I of England|beheading of King Charles I]] in 1649.&lt;ref&gt;Foxley (2013, p. 207), cited from the Wikipedia article on [[w:Levellers|Levellers]], accessed 2026-10-15.&lt;/ref&gt; However, that new government stopped granting as many monopolies and allowed so many commoners to become entrepreneurs that the GDP per capita began to grow at rates that are key features of Figure 1 for England and the 13 English colonies that became the US over a century later. 

One explanation for that economic growth was the diversity of independent publishers of pamphlets, books and newspapers. [[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]] of 1792&lt;ref&gt;Wikipedia &quot;[[w:Postal Service Act|Postal Service Act]]&quot;, accessed 2023-07-11.&lt;/ref&gt; made a major contribution to making the US what it is today.  Under that act, newspapers were delivered up to 100 miles for a penny, when postage for letters was between 6 and 25 cents depending on distance.&lt;ref&gt;Historian, USPS (2010).&lt;/ref&gt; McChesney and Nichols estimated that between 1840 and 1844, the US postal subsidy was a little over 0.2% of GDP.&lt;ref name=McC-N2010&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; 

During the first half of the nineteenth century the US had more independent newspaper publishers (more independent media organizations) per million population than at any other time or place in human history.&lt;ref&gt;John (1995). See also the interview with him at &quot;[[Media concentration per Columbia History Professor Richard John]]&quot; on Wikiversity, accessed 2026-10-12.&lt;/ref&gt; In 1932 with official unemployment over 20 percent, the conservative argument that blames the poor for their poverty did not sell newspapers, and the US still had sufficient competition in media markets that presidential candidate Franklin Roosevelt (FDR) got a mandate to spend the money to put humans back to work while taxing the ultra-wealthy like they had never been taxed before or since and during World War II to institute wage and price controls under the [[w:Office of Price Administration|Office of Price Administration]]. The latter prevented price gouging by government contractors that had generated inflation and stifled economic growth in previous major wars (World War I, the Civil War, and the War of 1812). Instead, the US averaged 7 percent per year growth for FDR's 12 years.&lt;ref&gt;The &quot;6 percent per year&quot; growth during FDR's 12 years was computed as the average of log(GDPpc) in the Maddison Data. To be precise, that computation produced 0.0597; converting that from a log scale to percent increase gives 0.0615. Using data from [[w:MeasuringWorth|MeasuringWorth]] instead of Maddison, we get 0.783 and 0.815, respectively. More documentation of this is available in &quot;[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]&quot; on Wikiversity, accessed 2026-10-13.&lt;/ref&gt; 

One of the most interesting features of Figure 1 is the behavior of the US economy in the administrations of [[w:Herbert Hoover|Herbert Hoover]] and FDR: Under Hoover, the economy declined at 9 percent per year for 4 years, a financial disaster that created the public demand for change that gave FDR the mandate to do what he did. Economic growth under two other presidents are numerically comparable to FDR's numbers but for much shorter periods: (1) [[w: Warren G. Harding|Warren G. Harding]] averaged 7 percent annual growth for two years before he died in 1923 during the [[w:Roaring Twenties|Roaring Twenties]]. (2) [[w:Rutherford B Hayes|Rutherford B Hayes]], who took office in 1877 at the end of the [[w:Panic of 1873|Panic of 1873]], oversaw 5.5 percent growth per year for his 4 years as president. Those impressive sounding average growth figures are so short that they are not noticeably different from other minor features visible in Figure 1. 

FDR also dramatically reduced inequality. For over a century between about 1820 and 1932 the top 1 percent of the US population owned roughly 45 percent of net personal wealth in the US, while the bottom 90 percent owned roughly 22 percent. Shortly after FDR died, the wealth of the top 1 percent and bottom 90 percent were even at roughly 30 percent. 

FDR also oversaw dramatic growth in the power of organized labor. Their successes continued to reduce inequality to roughly 22 percent for the top 1 percent vs. 37 percent for the bottom 90 percent when Ronald Reagan became president. Since then, [[w:Reaganomics|Reaganomics]] have been rebuilding the [[w:Robber baron (industrialist)|Robber Baron]]s.&lt;ref&gt;This history of inequality is documented in [[:File:Share of US wealth 90p99.svg|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023]] in the Wikiversity article on &quot;[[w:Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]&quot; in Wikiversity, accessed 2026-10-13.&lt;/ref&gt;

FDR made enemies among the wealthiest and most powerful. In a [[q:Franklin D. Roosevelt#Address at Madison Square Garden (1936)|speech in Maddison Square Garden in 1936]], FDR mentioned, &quot;the old enemies of peace: business and financial monopoly, speculation, reckless banking, class antagonism, sectionalism, war profiteering. ... They are unanimous in their hate for me -- and I welcome their hatred.&quot; They doubtless could see that they could no longer afford as many personal servants -- maids, butlers, cooks, chauffeurs, etc. -- as they had before FDR. 

Broadcasting was relatively new when FDR took office. The economics of broadcasting naturally favored a very few networks, whose power concerned FDR. In 1941 his [[w:Federal Communications Commission| Federal Communications Commission]] (FCC) adopted the [[w:Mayflower doctrine|Mayflower doctrine]], which prohibited broadcasters from advocacy. When FDR died, people with power were able to get the Mayflower doctrine replaced by the more lenient [[w:Fairness Doctrine|Fairness Doctrine]], which opened the floodgates for entrepreneurs of hate, who drove the [[w:McCarthyism|McCarthy Red Scare]] in the US and many other problems since. Over time this became worse, facilitated by increasing concentration of ownership of the media. That contributed to the end of the Fairness Doctrine starting in 1985. That revocation and especially the more recent rise of for-profit social media have amplified the ability of entrepreneurs of hate to make money both for commercial social media and for themselves by disseminating lies about others, e.g., immigrants.&lt;ref&gt;On how for-profit social media companies promote entrepreneurs of hate, see, e.g., DiResta (2026) and the Wikiversity article on, &quot;[[Managing the good and the evils of social media]]&quot;, accessed 2026-10-12. DiResta (2026, p. 1) described the promotion of &quot;influencers&quot; by for-profit social media as essentially unleashed &quot;a war for power, influence, and reality itself—a conflict in which the ends justify any means, and one that has ripped America apart, threatened our very democracy, and consumed my own life for the past decade, leading to death threats, congressional subpoenas, lawsuits, and legal bills running into seven figures.&quot; In the interview on Wikiversity, DiResta said her legal bills to that point had exceeded $7 million.&lt;/ref&gt; 

And the rate of economic growth in the US (and the Netherlands and the UK) have been slowing in recent decades, as is visible in Figure 1. Inequality expert [[w:Thomas Piketty|Piketty]] (2022, p. 139) noted, {{quote| 

In the United States, the national income per inhabitant rose at a rate ... of 2.2% between 1950 to 1990 when the top tax rate reached on average 72%.  The top rate was then cut in half, with the announced objective of boosting growth.  But in fact, growth fell by half, to 1.1% per annum between 1990 and 2020 ... . [A]ll the data at our disposal today suggest that virtually confiscatory tax rates have been an immense historical success. ...[T]he battle for equality and education ... has made economic development and human progress possible, and not the veneration of property, stability, and inequality.}}

Table 1 compares Piketty's growth numbers with computations based on the data from both Maddison and [[w:MeasuringWorth|MeasuringWorth]]: The numbers are close but not exact. 

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;&quot;
|+ Table 1. US growth rate 1950 to 1990 to 2020
|-
! 
! colspan=3 | US GDP per capita !! colspan=2 | Growth (% / yr)
|- 
! source || 1950 !! 1990 !! 2020 !! 1950 - 1990 !! 1990 - 2020 
|-
| Piketty || - || - || - || 2.2% || 1.1%  
|- 
| Maddison || $15,240 || $36,982 || $54,379 || 2.24% || 1.29%
|- 
| MeasuringWorth || $16,208 || $40,191 || $61,127 || 2.30% || 1.41%
|}

Acemoglu and Robinson (2012, pp.  105-106) provide a historical analysis supporting Piketty's concern about inequality slowing growth. They note that in 1588, England, Spain and France {{quote|

were ruled by relatively absolutist monarchs: [[w:Elizabeth I|Elizabeth I]] in England, [[w:Philip II of Spain|Philip II]] in Spain, and [[w:Henry III of France|Henry III]] in France. All were battling with assemblies of citizens -- such as the [[w:Parliament of England#Tudor era (1485–1603)|Parliament]] in England, the [[w:Cortes Generales#Habsburg rule (16th–17th centuries)|Cortes]] in Spain, and the [[w:Estates General (France)#Revival in 1560–1614|Estates-General]] in France -- that were demanding more rights and control over the monarchy. ... [T]he English Parliament and the Spanish Cortes had power over taxation, while the Estates-General did not. In Spain this mattered little, because after 1492 the Spanish Crown had a vast American empire and benefited massively from the gold and silver found there. In England ... Elizabeth I was far less financially independent, so she had to beg Parliament for more taxes. In exchange, Parliament demanded concessions, in particular restrictions of the right of Elizabeth to create monopolies. ... In Spain the Cortes lost a similar conflict.}} 

Consistent with Acemoglu and Johnson (2023) and the discussion above, inequality tends to increase elite control of the media, which it makes it harder for commoners to get a fair hearing of their concerns. 

=== Media conflicts of interest in IP law ===

Major media have [[w:Conflict of interest|conflicts of interest]] in honestly reporting on anything involving [[w:Intellectual property|intellectual property]] (IP) law. An example of Cardwell's law, mentioned above, is the [[w:Copyright Term Extension Act|Sonny Bono Copyright Term Extension Act]], also known as the Mickey Mouse Protection Act of 1998. The copyright term on ''[[w:Steamboat Willie|Steamboat Willie]]'', the first [[w:Mickey Mouse|Mickey Mouse]] animated film that Disney distributed, was extended multiple times by the US Congress following extensive lobbying by the film industry led by [[w:The Walt Disney Company|the Walt Disney Company]]. ''Steamboat Willie'', which debuted in 1928, finally entered the [[w:Public domain in the United States|US public domain]] on January 1, 2024. 

Eric Eldred and others filed suit, claiming that allowing any number of retroactive extensions violated the [[w:Copyright Clause|Copyright Clause]] of the US Constitution, which says, {{quote|

[the United States Congress shall have power] To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.}}

In ''[[w:Eldred v. Ashcroft|Eldred v. Ashcroft]]'' (2003), the [[w:Supreme Court of the United States|US Supreme Court]]  disagreed with Eldred and allowed the statute to remain in effect. However, that case drew enough public attention that copyright terms have not (yet) been extended further, and ''Steamboat Willie'' is currently in the pubic domain. 

[[w:Lawrence Lessig|Lessig]] (2004) ''[[w:Free Culture (book)|Free Culture]]'' describes other examples, where US copyright law seems to ''limit'' rather than &quot;promote the Progress of Science and useful Arts&quot;. This includes a case where a 2002 freshman, Jesse Jordan, modified a search engine to make it easier for other student to find pictures, research, notes, movie clips, etc., on an [[w:intranet|intranet]] at [[w:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] (RPI). The [[w:Recording Industry Association of America|Recording Industry Association of America]] (RIAA) sued Jordan and three other students for close to $100 billion for copyright violations -- six times the total profit of the film industry in 2001. Jordan had an uncle, who was an attorney, who explained that the legal bills for fighting that case would be at least $250,000. That would bankrupt his parents; they could not recover that money, even if they won. Jordan had saved $12,000 to cover part of his college expenses. RIAA accepted that and settled the case.&lt;ref&gt;Lessig (2004, p. 42).&lt;/ref&gt; 

This is an example of a [[w:Strategic lawsuit against public participation|strategic lawsuit against public participation]] (SLAPP), which in this case rested on questionable interpretations of copyright law. The major media have conflicts of interest in honestly reporting on anything involving (a) copyright laws, because much of their income is derived from copyright royalties, or (b) anything that might offend their major accounts, including major corporations. And anti-SLAPP laws, which exist in some jurisdictions but not others,&lt;ref&gt;[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel for the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]], supports anti-SLAPP laws and the &quot;JAWBONE act&quot;, to limit the power of the government to threaten questionable actions, like the Trump administration's threats against the ABC network over a Jimmy Kimmel monolog. See the Wikiversity article on &quot;[[Foundation for Individual Rights and Expression]]&quot;, accessed 2026-10-16.&lt;/ref&gt; limit the ability of major corporations to abuse their power against potential challengers without such deep pockets.

Similarly Graves (2023) asked, &quot;Do copyrights and paywalls on refereed academic journals violate the US constitution?&quot; No one expects to make money from copyright royalties for drafts submitted to refereed academic journals. They do it out of a felt need (a) to contribute to the future of humanity or (b) to build their reputation or (c) because promotions at many universities are based partly on publication records. Graves is not alone in his concern about this: Many European countries do not allow reports of research they fund to be published behind paywalls.&lt;ref&gt;Dimi Zagorski is concerned that copyright exceptions for scientific research are different in different European countries, and that's a problem for the Wikimedia Foundation. See the discussion of this in the Wikiversity article on &quot;[[Wikimedia concerns with European copyright rules including AI and scientific research]]&quot;. That's the opposite of US law, which grants patents to universities, research institutions, private companies, and individuals for innovations developed under government contracts, per the Bayh-Dole Act of 1980, described by Baker (2016, p. 80).&lt;/ref&gt;

Regarding patent law, the major media have conflicts of interest in discussing proposed changes affecting major accounts, but they make little use of patent law themselves. This seems to have contributed to the claim that, &quot;[O]ver the last four decades ... stronger patent protection seems to be associated with slower growth&quot; in the US, according to [[w:Dean Baker|Baker]] (2016, pp. 77-78). A potentially similar phenomenon appears in a comparison of 21 different European countries in an analysis that displays &quot;a positive relationship between anticorruption measures in a country and rate of innovation in that country as measured by patents per capita&quot;, discussed by Aghion et al. (2021, Figure 15.1b, pp. 292-293).&lt;ref&gt;Usher and Kim-Leffingwell (2022) documented how each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN), relatively small nonprofits independent of major media organizations and not funded by advertising, in a US federal court district was associated with on average 1.4 additional prosecutions for political corruption per year between 2003 and 2019. They found on average 10 federal prosecutions for political corruption for each federal court district per year. Thus, 1.4 additional prosecutions per year per district represents an increase of 14 percent per member of INN.&lt;/ref&gt;

=== Immigrants === 

We now return to discussing the initiation of economic growth following the beheading of Charles I in 1649 visible in Figure 1. Charles I was replaced by [[w:English Council of State|Lord Protector Cromwell]], who refused to authorize as many monopolies as his predecessors, as mentioned above. He also encouraged Jews and dissenting Christians to settle in England, as mentioned by Mokyr (2017, p. 53), cited above. 

But Mokyr is not alone in claiming that immigrants contribute to economic growth. Aghion et al. (2021, p. 266) cite Arkolakis et al. (2020) for explaining how the period 1880–1920 is noteworthy for two parallel evolutions. First, the United States caught up with England and France to become the most technologically advanced country in the world as well as the wealthiest in terms of GDP per capita. Second, the United States welcomed large numbers of immigrants, most notably from Europe. The purpose of the cited study was to demonstrate a causal connection between these two phenomena: in other words, to show that European immigration was responsible for the United States’ taking the lead in technology. ... More precisely, if the United States had completely closed its borders to all foreign immigrants in 1880, its per capita GDP would have been 30 percent lower than the level actually attained in 1920&quot;. 

=== Industrial Revolution ===

There seems to be a broad consensus that the [[w:Industrial Revolution|Industrial Revolution]] began in England. However, there is less agreement on when it began. 

Mokyr (2017, p.  3) says, &quot;The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society.&quot; He described &quot;three cultural elements&quot; required for &quot;sustainable economic growth propelled by technological progress&quot;: (1, 2) The belief that it was both ''possible'' and ''desirable''. (3) Supportive policy measures had to be implemented. &quot;[B]efore 1750 they did not produce anything like an Industrial Revolution or sustainable economic growth propelled by technological progress.&lt;ref&gt;Mokyr (2017, pp. 19-20).&lt;/ref&gt; Acemoglu (2026, p. 53) similarly mentioned the eighteenth century in saying, &quot;The breakthroughs in production technology that started in the middle of the eighteenth century in Britain unleashed the Industrial Revolution.&quot; 

Aghion et al. (2021, esp. pp. 29-30) offer a ''later'' date for the start of the Industrial Revolution: They report that, &quot;European economies prior to 1820 experienced a series of phases of growth and decline. This raises a question: How did the pre-1820 period differ from the takeoff that started with the first Industrial Revolution?&quot; Aghion et al. (2021, p. 20) describe the &quot;first Industrial Revolution&quot; as having been triggered by &quot;the invention of the steam engine by James Watt in the 1770s ... . The invention of electricity set off the second Industrial Revolution, whose golden age was in the first half of the twentieth century. This second revolution originated with the invention of the [[w:Incandescent light bulb|light bulb]] by [[w:Thomas Edison|Thomas Edison]] in 1879 and of the [[w:dynamo|dynamo]] by [[w:Werner von Siemens|Werner von Siemens]] in 1866.&quot;&lt;ref&gt;The dynamo was independently invented by [[w:Henry Wilde (engineer)|Henry Wilde]], [[w:Charles Wheatstone|Charles Wheatstone]], [[w:Samuel Alfred Varley|Samuel Alfred Varley]] and Siemens, all about the same time, according to the Wikipedia article on [[w:Dynamo#Practical designs|dynamo]], accessed 2026-10-18.&lt;/ref&gt;

The image in Figure 1 raises questions about these claims by Mokyr, Acemogu, and Aghion et al.: It shows growth followed by stagnation in Holland but not &quot;phases of growth and decline&quot; with the exception of wars and [[w:Herbert Hoover|Herbert Hoover]]. It also does ''not'' show the slope (i.e., rate of growth) increasing starting around 1750 or the 1770s or 1879 or 1886. Instead it shows sustained growth in GDP per capita for England / Great Britain / the UK starting ''abruptly'' after 1649 with economic disruptions during the reign of [[w:Anne, Queen of Great Britain|Queen Anne]] (1702-1714) and the [[w:War of the Spanish Succession|War of the Spanish Succession]] (1701-1714). This was followed by relatively steady but slower growth as England was ruled by Kings [[w:House of Hanover#Monarchs of Great Britain, Ireland, and Hanover|George I, II, and III]] (1714-1820). Another dramatic increase in the slope started abruptly about 1819. 

Acemoglu and Johnson (2023, chapter 1) are more consistent with Figure 1. They said that, &quot;the Industrial Revolution ... [was] rooted in institutional changes that started empowering the middling sort of English people from the sixteenth and seventeenth centuries onward.&quot; ... However, &quot;the quality of life of a European peasant circa 1700 was not much different from that of an Egyptian peasant two thousand or seven thousand years previously.&quot; (chapter 5) They also claim that, &quot;English national production per capita doubled from 1500 to 1700&quot;, (chapter 5), and &quot;Growth emerged slowly in the second half of the eighteenth century as the Industrial Revolution, based on major technological breakthroughs and their application in industry, took root.&quot; (p. 45, chapter 2)

Indeed, Figure 1 shows English GDP per capita doubling between 1500 and 1700, but it appears to have been fairly constant from 1500 to 1649, then roughly doubled in the next half century. [The data on England up to 1700 in Figure 1 came from Broadberry et al. (2015), which was not available when Acemoglu and Robinson were writing ''Why Nations Fail'', which appeared in 2012.] 

Acemoglu and Robinson (2012, ch. 11) describe the financial instability and slow growth during the reigns of Queen Anne and George I, II, and III. After the Glorious Revolution, France, Spain, the pope, and others refused to recognize [[w:William III of England|William and Mary]] (1689-1702) and then Queen Anne as heads of state of England. Instead they continued to support the claims of James II and then his son, [[w:James Francis Edward Stuart|James Francis Edward Stuart]] (the &quot;Old Pretender&quot;, a brother of Queen Anne), after James II died in 1701. During the War of the Spanish Succession, each side tried to exploit the internal divisions of the other. Owen (1938, ch. 8) said France almost succeeded in an attempt in 1708 to [[w:Planned French invasion of Britain (1708)|land between 5 and 6 thousand soldiers in northeast Scotland]]. [[w:Louis XIV|Louis XIV]] had hoped to get the Old Protector named as King James VIII of Scotland. There were still [[w:Jacobitism|Jacobites]] in Scotland, who supported the Old Pretender and likely would have supported the invaders if they had landed. If they had succeeded, it would have created major problems for Queen Anne and her supporters. However, the Royal Navy prevented the invaders from disembarking, and they returned to France. 

When Queen Anne died in 1702, she was succeeded by George I. His Parliament was dominated by the Whig party, which represented the new mercantile and economic interests and had been the main organization behind the Glorious Revolution. Once in power, they were tempted to use their newly found position to prey on the rights of others but were constrained both by competing groups in Parliament, &quot;and by the very institutions that they had fought to introduce to strengthen Parliament and to prevent the emergence of a new absolutism and the return of the Stuarts&quot;,&lt;ref&gt;After James II fled England, the new government worked to block [[w:Jacobitism|Jacobitism]], which attempted to restore the [[w:House of Stuart|House of Stuart]] to the thrones of Scotland and Britain. When James II died, his son became titular head of this movement, who became known as &quot;the [[w:James Francis Edward Stuart|Old Pretender]]&quot;. After he died, he was replaced by his son, who became known as &quot;the [[w:Charles Edward Stuart|Young Pretender]]&quot;.&lt;/ref&gt; according to Acemoglu and Robinson (2012, pp. 302-303). The Whigs gave &quot;[[w:Common land|common land]]&quot; to nobles, confiscating it from commoners. Some commoners, who had traditionally shared that land, joined &quot;Blacks&quot;, so called, because they blackened their faces when hunting on former commons or destroying the property of the nobility or extorting money from nobles or their employees. Parliament retaliated by passing the [[w:Black Act 1723|Black Act 1723]], under which someone accused of [[w:Poaching|poaching]] a [[w:Hare|hare]], for example, could under certain circumstances receive a summary judgment of guilty and then be executed without delay if apprehended later.&lt;ref&gt;Newman and Brown, eds. (1997, p. 58). See also the Wikipedia article on &quot;[[w:Black Act 1723|Black Act 1723]]&quot;, accessed 2026-10-17.&lt;/ref&gt; 

But not all accused were summarily executed. In 1724 &quot;John Huntridge was accused of aiding deer stealers and abetting known Blacks, both crimes punishable by hanging. The prosecution of Huntridge came right from the top, initiated by the Regency Council of Lords Justices ... . After a trial of eight or nine hours, the jury found Huntridge innocent, partly on procedural grounds, since there were irregularities with the way the evidence had been collected. ... The jury was made up not of Huntridge’s peers, but of major landowners and gentry, who ought to have sympathized&quot; with the noble, whose property had been attacked by Blacks allegedly aided by Huntridge.&lt;ref&gt;Acemoglu and Robinson (2012, pp. 304-305).&lt;/ref&gt; 

These kinds of actions by the Parliaments of George I, II, and III might explain the slower growth during that period compared to that of the last half of the seventeenth century. That slower growth ended abruptly at the time of the [[w:Peterloo massacre|Peterloo Massacre]], 1819-08-16, visible in Figure 1. The growth of factories implied growth of cities, where workers lived closer together and could more easily organize. A campaign arose with multiple demands: They opposed the [[w:Corn Laws|Corn Laws]], which kept prices high for cereal grains and products made from them. They also demanded &quot;universal [adult male] suffrage&quot; and secret ballots in elections. A series of public meetings discussing these grievances led to one 1819-08-16, which attracted roughly 60,000. Acemoglu and Robinson (2012, p. 207) reported that, {{quote|

The authorities were very nervous about the meeting, and a force of six hundred cavalry of the [[w:15th The King's Hussars|Fifteenth Hussars]] had been assembled. As the speeches began, a local magistrate decided to issue a warrant for the arrest of the speakers. As police tried to enforce the warrant, they met with the opposition of the crowd, and fighting broke out. At this point the Hussars charged the crowd. Within a few chaotic minutes, eleven people were dead and probably six hundred wounded. The ''[[w:Manchester Observer|Manchester Observer]]'' called it the Peterloo Massacre. But given the changes that had already taken place in economic and political institutions, long-run repression was not a solution in England. The Peterloo Massacre would remain an isolated incident. Following the riot, the political institutions in England gave way to the pressure, and the destabilizing threat of much wider social unrest, particularly after the 1830 revolution in France against [[w:Charles X of France|Charles X]], who had tried to restore the absolutism destroyed by the French Revolution of 1789. In 1832 the government passed the [[w:Reform Act 1832|First Reform Act]]. It enfranchised [[w:Birmingham|Birmingham]], [[w:Leeds|Leeds]], [[w:Manchester|Manchester]], and [[w:Sheffield|Sheffield]], and broadened the base of voting so that manufacturers could be represented in Parliament. ... [I]n 1846 they managed to get the hated Corn Laws repealed ... .}}

The ''Manchester Observer'' closed roughly two years later after [[w:Manchester Observer#Closure by repeated prosecution|repeated prosecutions]] for [[w:Seditious libel|seditious libel]]. However, the British government placed enough other limits on its power that enough other commoners were allowed to become successful entrepreneurs that the rate of economic growth in the UK increased immediately, as indicated by the increase in the slope of the UK line in Figure 1 until World Wars I and II. 

== In sum ==

The above suggests that the rate of economic growth changes with either (a) a change in administration -- a new head of state -- or (b) some other major event like war, the Peterloo Massacre, or the [[w:Wall Street crash of 1929|Wall Street crash of 1929]]. Policy changes that can increase growth include (1) limits on grants of monopolies, (2) diverse ownership and control of the major media and (3) support for immigrants. 

However, correlation does not imply causation. [[w:Randomized controlled trial|Randomized controlled trial]]s have made important contributions to human understanding of economic growth, as documented by Banerjee, Duflo, and Kramer, cited above and recommended in the Wikiversity article on &quot;[[Information is a public good: Designing experiments to improve government]]&quot;. Failing that, more analyses of data from different political jurisdictions with different media policies could be useful, as described by Neff and Pickard (2024).&lt;ref&gt;Graves (2025) includes an analysis of the data in Neff and Pickard (2024).&lt;/ref&gt; 

== Caveats ==

We should be more confident of the results of randomized controlled trials, as recommended by Banerjee, Duflo, and Kremer, than of anyone interpreting the historical record, even by Nobel economists. 

The above attempts to accurately reflect the perspectives of all sources cited. Readers are invited to revise this article, subject to the standard Wikimedia rules of writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”{{'}}''' page]] -- and treating others with respect. 

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!-- Daron Acemoglu (2026-08-11) What Happened to Liberal Democracy?: Remaking a Politics of Shared Prosperity (Function)--&gt;{{cite Q|Q141241291}}
* &lt;!--Daron Acemoglu and Simon Johnson (2023) Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity (PublicAffairs)--&gt;{{cite Q|Q125292212}}
* &lt;!--Daron Acemoglu and James A. Robinson (2012) Why Nations Fail (Crown)--&gt;{{cite Q|Q7997840}} 
* &lt;!--Philippe Aghion, Céline Antonin, and Simon Bunel (2021) The Power of Creative Destruction: Economic Upheaval and the Wealth of Nations (Harvard University Press)--&gt;{{cite Q|Q137641358}}
* &lt;!--Ajzenman, Nicolas (29 January 2023). &quot;Migrants don't cause crime rates to increase — but false perceptions endure anyway&quot;. The Conversation.--&gt;{{cite Q|Q141758664}}
* &lt;!--Samuel Alexander (2026-07) &quot;Game Theory and the Dilemma of Degrowth: Why Embrace Degrowth if Other States or Societies Do Not?&quot;--&gt;{{cite Q|Q141968019}}
* &lt;!--Costas Arkolakis, Michael Peters, and Sun Kyoung Lee, (2020) &quot;European Immigrants and the United States’ Rise to the Technological Frontier&quot;--&gt;{{cite Q|Q141678287}}
* &lt;!--Dean Baker (2016) Rigged: How globalization and the rules to the modern economy were structured to make the rich richer--&gt;{{cite Q|Q100216001}}
* &lt;!--Abhijit V. Banerjee and Esther Duflo (2019) Good Economics for Hard Times (PublicAffairs)--&gt;{{cite Q|Q85764011}}
* &lt;!--Jutta Bolt and Jan Luiten van Zanden (2024-09-18) &quot;Maddison style estimates of the evolution of the world economy: A new 2023 update&quot;, Journal of Economic Surveys, 1-41--&gt;{{cite Q|Q126723821}}
* &lt;!--S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)--&gt;{{cite Q|Q57945943}}
* &lt;!--Donald S. L. Cardwell (1972) Turning Points in Western Technology (Neale Watson, Science History Pubs)--&gt;{{cite Q|Q141795083}}
* &lt;!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition--&gt;{{cite Q|Q141156783}}
* &lt;!--Renée DiResta and Rachel Kleinfeld (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies--&gt;{{cite Q|Q141158362}}
* &lt;!--Katarzyna Dośpiał-Borysiak (2017) &quot;Model of state management of petroleum sector -- Case of Norway--&gt;{{cite Q|Q141774179}}
* &lt;!--Rachel Foxley (2013) The Levellers: Radical Political Thought in the English Revolution (Oxford University Press)--&gt;{{cite Q|Q136227988}}
* &lt;!--Benjamin Gonzalez O’Brien, Loren Collingwood, Stephen Omar El-Khatib (2017-05-07) &quot;The Politics of Refuge: Sanctuary Cities, Crime, and Undocumented Immigration--&gt;{{cite Q|Q55932800}}
* &lt;!--Graves (2025-01-08) Defend Free Speech For All (Town Hall)--&gt;{{cite Q|Q132855526}}
* &lt;!--Graves (2023-03-21)  Do copyrights and paywalls on academic journals violate the US Constitution?--&gt;{{cite Q|Q126116187}}
* &lt;!--Christopher Hibbert, Ben Weinreb, John Keay, Julia Keay, and Matthew Weinreb (2011-09-09) The London Encyclopaedia, 3rd ed.--&gt;{{cite Q|Q137188993}}
* &lt;!-- Historian, USPS (2010) Postage Rates for Periodicals --&gt;{{cite Q|Q112167199}}
* &lt;!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse--&gt;{{cite Q|Q54641943}}
* &lt;!--Volker Leppin and Timothy J. Wengert (2015) &quot;Sources for and against the Posting of the Ninety-Five Theses&quot;--&gt;{{cite Q|Q141737347}}
* &lt;!--Lawrence Lessig (2004) Free Culture--&gt;{{cite Q|Q830526}}
* &lt;!--Olivier Marie and Paolo Pinotti (2024-02-01) &quot;Immigration and Crime: An International Perspective&quot;, Journal of Economic Perspectives. 38 (1): 181–200--&gt;{{cite Q|Q128434874}}
* &lt;!--Martínez, Daniel E.; Martínez-Schuldt, Ricardo D.; Cantor, Guillermo (2017). &quot;
Providing Sanctuary or Fostering Crime? A Review of the Research on 'Sanctuary Cities' and Crime&quot;
. Sociology Compass. 12 e12547. doi:10.1111/soc4.12547. ISSN 1751-9020.--&gt;{{cite Q|Q55932802}}
* &lt;!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) --&gt;{{cite Q|Q104888067}}.
* &lt;!-- Robert W. McChesney; John Nichols (2016). People get ready--&gt;{{cite Q|Q87619174}}
* &lt;!--Robert Michels (1911)  Les Partis politiques, essai sur les tendances oligarchiques des démocraties--&gt;{{cite Q|Q7210216}}
* &lt;!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Princeton University Press)--&gt;{{cite Q|Q136118590}}
* &lt;!--Joel Mokyr (1994) &quot;Cardwell's Law and the political economy of technological progress&quot;, Research Policy 23(5)561-574--&gt;{{cite Q|Q139772607}}
* &lt;!--Ashley N. Muchow &amp; Taisiia Stanishevska (2026) &quot;From Headlines to Handcuffs: Local News Coverage of Immigrant Crime and Hispanic Arrest&quot;, Justice Quarterly, vol. 43(3) (https://doi.org/10.1080/07418825.2025.2550307)--&gt;{{cite Q|Q141760694}}
* &lt;!--Neff and Pickard (2024) &quot;Funding Democracy: Public Media and Democratic Health in 33 Countries&quot;--&gt;{{cite Q|Q131468289}}
* &lt;!--Gerald G. Newman and Leslie Ellen Brown, eds. (1997) Britain in the Hanoverian Age, 1714–1837: An Encyclopedia--&gt;{{cite Q|Q141759196|authors = Gerald G. Newman and Leslie Ellen Brown, eds.}}
* &lt;!--John Hely Owen (1938)  War at Sea Under Queen Anne 1702–1708--&gt;{{cite Q|Q141757101}}
* &lt;!--John Perkins (2004) Confessions of an Economic Hit Man--&gt;{{cite Q|Q400098}}
* &lt;!--Andrew Pettegree (2015-10-27) Brand Luther: How an Unheralded Monk Turned His Small Town into a Center of Publishing, Made Himself the Most Famous Man in Europe--and Started the Protestant Reformation--&gt;{{cite Q|Q141761223}}
* &lt;!--Thomas Piketty (2022) A Brief History of Equality--&gt;{{cite Q|Q139886018}}
* &lt;!--Joad Raymond (2006-11-02) Pamphlets and Pamphleteering in Early Modern Britain--&gt;{{cite Q|Q141761849}}
* &lt;!--James A. Robinson (2025-04) &quot;Paths to the periphery&quot;, NBER Working Paper 33671--&gt;{{cite Q|Q141967690}}
* &lt;!--Nick Roll (2017-07-16) &quot;Correcting Jeff Sessions: Attorney general railed against crime in sanctuary cities, citing two professors’ work. But his claims are contrary to their conclusions.--&gt;{{cite Q|Q141735231}}
* &lt;!--J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)--&gt;{{cite Q|Q137669960}}
* &lt;!--I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)--&gt;{{cite Q|Q137669987}}
* &lt;!--Kelly Swanson (2017-09-17) &quot;The Trump administration rejects its own study finding refugees help the economy--&gt;{{cite Q|Q141900700}}
* &lt;!--Ted Trainer (2025-03) &quot;The 2024 Nobel Prize for Economics: An example of how conventional economics misrepresents reality--&gt;{{cite Q|Q142065447}}
* &lt;!--Union of Concerned Scientists (2019-01-11) &quot;The Trump Administration Manipulates Study on Economic Benefits of Refugees in US&quot;--&gt;{{cite Q|Q141898260|author=Union of Concerned Scientists}}
* &lt;!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption --&gt;{{Cite Q|Q134715465}}
* &lt;!--J. L. van Zanden  and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30--&gt;{{cite Q|Q137670058}} 
* &lt;!--Brittney Washington (2018-06-19) &quot; A Pamphlet War in England, 1641-1643&quot;--&gt;{{cite Q|Q141822147}}
* &lt;!--Tom K. Wong (2017-01-26) &quot;The Effects of Sanctuary Policies on Crime and the Economy&quot;--&gt;{{cite Q|Q98775875}}

[[Category:Original research]]
[[Category:Research]]
[[Category:Economics]]
[[Category:Macroeconomics]]
[[Category:Gross domestic product]]
[[Category:Economic growth]]
[[Category:Modern Monetary Theory]]
[[Category:Political science]]
[[Category:Freedom and abundance]]
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'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. Before he starts to do it, he must make sure that his intentions are correct. Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions. Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

[[Category:Islamic Studies]]
[[Category:Atcovi's Work]]</text>
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'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. '''Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah'''. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). '''He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions.''' Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

[[Category:Islamic Studies]]
[[Category:Atcovi's Work]]</text>
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'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. '''Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah'''. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). '''He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions.''' Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad Nasir al-Din al-Albani{{efn|{{langx|ar|مُحَمَّد نَاصِر ٱلدِّيْن الْأَلْبَانِي}}; {{langx|sq|Muhamed Nazir ed-Din}}}} ==

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

[[Category:Islamic Studies]]
[[Category:Atcovi's Work]]</text>
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'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. '''Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah'''. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). '''He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions.''' Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad Nasir al-Din al-Albani ==

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

[[Category:Islamic Studies]]
[[Category:Atcovi's Work]]</text>
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'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting the Murji'ah ===
&lt;small&gt;Imam Abdulaziz ibn Baz was asked: &quot;Some people claim that those who go to graves and call upon their occupants are committing an act of shirk, but they themselves are not mushrikeen. Is this correct?&quot;&lt;/small&gt;

* &lt;u&gt;Answer:&lt;/u&gt; '''No, this is not correct. Those who go to graves, call upon their occupants besides Allah, and seek their aid besides Allah are committing major shirk.''' This is the very same practice of the kuffar of Quraysh and others who worshipped al Lat, al Uzza, and similar idols. This is major shirk. Therefore, it is obligatory to warn them of this, admonish them, and call upon them to repent to Allah from their shirk and renew their Islam.
** SOURCE: [https://binbaz.org.sa/fatwas/20209/%D8%AF%D8%B9%D8%A7%D8%A1-%D8%A7%D8%B5%D8%AD%D8%A7%D8%A8-%D8%A7%D9%84%D9%82%D8%A8%D9%88%D8%B1-%D8%B4%D8%B1%D9%83-%D8%A7%D9%83%D8%A8%D8%B1 1]

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. '''Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah'''. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). '''He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions.''' Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad Nasir al-Din al-Albani ==

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

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      <text bytes="17527" sha1="i1hc7brwgl2s6xl34oz7v1u9ivepkpd" xml:space="preserve">{{displaytitle|The Sayings of the Scholars &amp; the Salaf}}
'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting the Murji'ah ===
&lt;small&gt;Imam Abdulaziz ibn Baz was asked: &quot;Some people claim that those who go to graves and call upon their occupants are committing an act of shirk, but they themselves are not mushrikeen. Is this correct?&quot;&lt;/small&gt;

* &lt;u&gt;Answer:&lt;/u&gt; '''No, this is not correct. Those who go to graves, call upon their occupants besides Allah, and seek their aid besides Allah are committing major shirk.''' This is the very same practice of the kuffar of Quraysh and others who worshipped al Lat, al Uzza, and similar idols. This is major shirk. Therefore, it is obligatory to warn them of this, admonish them, and call upon them to repent to Allah from their shirk and renew their Islam.
** SOURCE: [https://binbaz.org.sa/fatwas/20209/%D8%AF%D8%B9%D8%A7%D8%A1-%D8%A7%D8%B5%D8%AD%D8%A7%D8%A8-%D8%A7%D9%84%D9%82%D8%A8%D9%88%D8%B1-%D8%B4%D8%B1%D9%83-%D8%A7%D9%83%D8%A8%D8%B1 1]

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]
*O servants of Allah! Indeed, it does not harm us if our clothes become torn, nor does it harm us if we go hungry. '''What harms us is that our religion should be taken away from us in exchange for some worldly gain. Therefore, we Muslims must return to Allah.'''
**al-Fawakih al-Janiyyah fi al-Khutab wa al-Muhadarat al-Saniyyah, p. 149. 

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

* '''The more merciful a person is towards people, the more merciful Allah is towards him, because the reward is of the same kind as the deed.''' So if a person finds harshness in his heart towards the believing servants of Allah, he must treat this disease.
** Al-Ta'liq 'ala al-Muntaqa (4/121) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* '''When you retire to your bed to sleep, recite [Ayat al-Kursi], for there will remain a protector over you from Allah, and no devil will come near you until morning'''. These are a few simple words that will protect you.
** Sharh Riyad al-Salihin (4/690) [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*Sinful deeds sow the seeds of other sins and give birth to one another, until it becomes difficult for a servant to abandon them and break free from them.
**Al-Jawab al-Kafi (p. 143) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*Patience is a means of attaining every form of perfection. The most perfect of creation are those who have the greatest patience, and no one falls short of attaining the perfection that is possible for him except due to the weakness of his patience.
**Tariq al-Hijratayn (2/578) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. '''Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah'''. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). '''He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions.''' Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Rajab al-Hanbali ==

* '''Allah loves that His servants ask Him for all their religious and worldly needs, like their food, drink, clothing, etc.,''' just as they ask Him for His guidance and mercy, because if the servant asks for everything he needs from Allah, he has shown his need of Him, and Allah loves that.
** Jami al-Ulum wa al-Hikam, vol. 2, p. 38.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad Nasir al-Din al-Albani ==

* What matters is not doing a lot of worship. What matters is that the worship follows the Sunnah and stays far from innovation. Ibn Mas'ud, may Allah be pleased with him, pointed to this when he said: &quot;Doing a little while following the Sunnah is better than doing a lot while following an innovation.&quot;
** Al-Silsilah al-Sahihah (5/14), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

== Abu Bakr al-Marrudhi ==

* I heard Abu Abdillah Ahmad ibn Hanbal, may Allah have mercy upon him, say: &quot;[My wife] Umm Salih was with me for 30 years, and we never disagreed in a single word.&quot;
** Tarikh Baghdad (14/438), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibrahim al-Taymi ==

* Two things have cut me off from the pleasure of this world: remembering death, and remembering standing before Allah the Exalted.
** Al-Tadhkirah, p. 125, [Al-Baseerah Islamic Center in Plainfield, New Jersey]

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'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting the Murji'ah ===
&lt;small&gt;Imam Abdulaziz ibn Baz was asked: &quot;Some people claim that those who go to graves and call upon their occupants are committing an act of shirk, but they themselves are not mushrikeen. Is this correct?&quot;&lt;/small&gt;

* &lt;u&gt;Answer:&lt;/u&gt; '''No, this is not correct. Those who go to graves, call upon their occupants besides Allah, and seek their aid besides Allah are committing major shirk.''' This is the very same practice of the kuffar of Quraysh and others who worshipped al Lat, al Uzza, and similar idols. This is major shirk. Therefore, it is obligatory to warn them of this, admonish them, and call upon them to repent to Allah from their shirk and renew their Islam.
** SOURCE: [https://binbaz.org.sa/fatwas/20209/%D8%AF%D8%B9%D8%A7%D8%A1-%D8%A7%D8%B5%D8%AD%D8%A7%D8%A8-%D8%A7%D9%84%D9%82%D8%A8%D9%88%D8%B1-%D8%B4%D8%B1%D9%83-%D8%A7%D9%83%D8%A8%D8%B1 1]

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]
*O servants of Allah! Indeed, it does not harm us if our clothes become torn, nor does it harm us if we go hungry. '''What harms us is that our religion should be taken away from us in exchange for some worldly gain. Therefore, we Muslims must return to Allah.'''
**al-Fawakih al-Janiyyah fi al-Khutab wa al-Muhadarat al-Saniyyah, p. 149. 

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

* '''The more merciful a person is towards people, the more merciful Allah is towards him, because the reward is of the same kind as the deed.''' So if a person finds harshness in his heart towards the believing servants of Allah, he must treat this disease.
** Al-Ta'liq 'ala al-Muntaqa (4/121) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* '''When you retire to your bed to sleep, recite [Ayat al-Kursi], for there will remain a protector over you from Allah, and no devil will come near you until morning'''. These are a few simple words that will protect you.
** Sharh Riyad al-Salihin (4/690) [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*Sinful deeds sow the seeds of other sins and give birth to one another, until it becomes difficult for a servant to abandon them and break free from them.
**Al-Jawab al-Kafi (p. 143) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*Patience is a means of attaining every form of perfection. The most perfect of creation are those who have the greatest patience, and no one falls short of attaining the perfection that is possible for him except due to the weakness of his patience.
**Tariq al-Hijratayn (2/578) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. '''Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah'''. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). '''He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions.''' Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Rajab al-Hanbali ==

* '''Allah loves that His servants ask Him for all their religious and worldly needs, like their food, drink, clothing, etc.,''' just as they ask Him for His guidance and mercy, because if the servant asks for everything he needs from Allah, he has shown his need of Him, and Allah loves that.
** Jami al-Ulum wa al-Hikam, vol. 2, p. 38.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad Nasir al-Din al-Albani ==

* What matters is not doing a lot of worship. What matters is that the worship follows the Sunnah and stays far from innovation. Ibn Mas'ud, may Allah be pleased with him, pointed to this when he said: &quot;Doing a little while following the Sunnah is better than doing a lot while following an innovation.&quot;
** Al-Silsilah al-Sahihah (5/14), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

== Abu Bakr al-Marrudhi ==

* I heard Abu Abdillah Ahmad ibn Hanbal, may Allah have mercy upon him, say: &quot;[My wife] Umm Salih was with me for 30 years, and we never disagreed in a single word.&quot;
** Tarikh Baghdad (14/438), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibrahim al-Taymi ==

* Two things have cut me off from the pleasure of this world: remembering death, and remembering standing before Allah the Exalted.
** Al-Tadhkirah, p. 125, [Al-Baseerah Islamic Center in Plainfield, New Jersey]

[[Category:Islamic Studies]]
[[Category:Atcovi's Work]]</text>
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      <text bytes="18999" sha1="2m2acm3dr97qjmssctk5dxap9pnk1sc" xml:space="preserve">{{displaytitle|The Sayings of the Scholars &amp; the Salaf}}
'''Aaqib Azeez'''

&lt;big&gt;بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ&lt;/big&gt;&lt;br&gt;
''In the Name of Allah, the Beneficent, the Merciful''

&lt;big&gt;إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ&lt;/big&gt;&lt;br&gt;
''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:''

Allaah, the Exalted, says:

* {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]]
* {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]]
* {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]]

This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf.

== Abdulaziz ibn Abdullah Al Baz ==

* '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237)

=== Refuting the Murji'ah ===
&lt;small&gt;Imam Abdulaziz ibn Baz was asked: &quot;Some people claim that those who go to graves and call upon their occupants are committing an act of shirk, but they themselves are not mushrikeen. Is this correct?&quot;&lt;/small&gt;

* &lt;u&gt;Answer:&lt;/u&gt; '''No, this is not correct. Those who go to graves, call upon their occupants besides Allah, and seek their aid besides Allah are committing major shirk.''' This is the very same practice of the kuffar of Quraysh and others who worshipped al Lat, al Uzza, and similar idols. This is major shirk. Therefore, it is obligatory to warn them of this, admonish them, and call upon them to repent to Allah from their shirk and renew their Islam.
** SOURCE: [https://binbaz.org.sa/fatwas/20209/%D8%AF%D8%B9%D8%A7%D8%A1-%D8%A7%D8%B5%D8%AD%D8%A7%D8%A8-%D8%A7%D9%84%D9%82%D8%A8%D9%88%D8%B1-%D8%B4%D8%B1%D9%83-%D8%A7%D9%83%D8%A8%D8%B1 1]

=== Refuting Terrorism ===
*&quot;From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''&quot;
**SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1]

== [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] ==

* &quot;'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: &quot;'Abd Allah b. Yusuf narrated to us: Malik narrated to us...&quot; or I read ''Imam'' Muslim say: &quot;Yahya b. Yahya narrated to us....&quot; I forget all the concerns of the ''dunya'' (world) and its problems&quot;
** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1]
*O servants of Allah! Indeed, it does not harm us if our clothes become torn, nor does it harm us if we go hungry. '''What harms us is that our religion should be taken away from us in exchange for some worldly gain. Therefore, we Muslims must return to Allah.'''
**al-Fawakih al-Janiyyah fi al-Khutab wa al-Muhadarat al-Saniyyah, p. 149. 

=== Refuting Osama bin Laden ===

* The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&amp;sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.'''
** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1]

==Salih al-Fawzan==
*Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.'''
**SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1]

== Muhammad ibn Sāliḥ al-Uthaymin ==

* '''The more merciful a person is towards people, the more merciful Allah is towards him, because the reward is of the same kind as the deed.''' So if a person finds harshness in his heart towards the believing servants of Allah, he must treat this disease.
** Al-Ta'liq 'ala al-Muntaqa (4/121) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* '''When you retire to your bed to sleep, recite [Ayat al-Kursi], for there will remain a protector over you from Allah, and no devil will come near you until morning'''. These are a few simple words that will protect you.
** Sharh Riyad al-Salihin (4/690) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*''&quot;He who lets the people hear of his good deeds intentionally...&quot;'' A person says words that seem to draw him closer to Allah, but only to be heard by people and praised by them. Hence, Allah lets his real intention become known; then the people begin talking about him. ''&quot;And the person who does good things in public to show off...&quot;'' - actions are seen and words are heard. A person either says or does something. So whoever says something so people can hear about it and does something so people can see it, Allah will make his condition known. For this reason, there is a warning against showing off and seeking fame in this hadeeth. '''Question''': Sometimes these feelings befall a person, or come to his mind and he isn't able to repel these thoughts and feelings. '''Answer:''' This is correct. Sometimes the idea of doing things to be seen or heard occurs to an individual and he can't repel them. '''The cure for this is to say to yourself, &quot;I'm doing this so people can follow me, not praise me.&quot; If you do something so others can take you as an example, showing off vanishes. Then you should feel the burden of responsibility: You are now a leader and you want people to follow you in goodness.''' If you obey Shaytaan in his speech, &quot;you are showing off; you aren't doing this act for Allah,&quot; you will turn away from this act and your soul will feel the things Shaytaan whispered to you. Allah's aide is sought!
**The Book of Heart Softeners, page 63-64.

== Muhammad ibn Abd al-Wahhab ==

* Abu Hurairah asked the Messenger (ﷺ): &quot;Who will be the most fortunate and be granted your intercession?&quot; The Messenger said: &quot;Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'&quot; This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah.
** SOURCE: ''Book of Monotheism'', page 27.

== Ibn Qayyim al-Jawziyya ==

* The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path.
** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
* I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: &quot;'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''.&quot;
** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*Sinful deeds sow the seeds of other sins and give birth to one another, until it becomes difficult for a servant to abandon them and break free from them.
**Al-Jawab al-Kafi (p. 143) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*Patience is a means of attaining every form of perfection. The most perfect of creation are those who have the greatest patience, and no one falls short of attaining the perfection that is possible for him except due to the weakness of his patience.
**Tariq al-Hijratayn (2/578) [Al-Baseerah Islamic Center in Plainfield, New Jersey]
*A scholar said, &quot;To have patience means that one's common sense and religious motives are stronger than one's whims and desires.&quot; It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''.
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15.
*Every step of the way, man needs patience in carrying out an act of worship. '''Before he starts to do it, he must make sure that his intentions are correct. He should check his sincerity, and seek to avoid showing off in performing any act of worship. Whilst he is performing any act of worship, he must try to perfect it, to keep his intention pure and his mind focused on the purpose of performing that act of worship, namely that it is to please Allah'''. After completing an act of worship, he must abstain from doing anything that could corrupt his worship. Allah told us: ''O you who believe! Cancel not your charity by reminders of your generosity or by injury'' (Al-Baqarah 2:264). '''He should exercise patience in refraining from admiring and feeling proud of his performance, as this is more damaging than committing many other, more visible, wrong actions.''' Similarly, he should always be discreet and refrain from telling others about his acts of worship. 
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 39.
*'Umar ibn al-Khattab, may Allah be pleased with him, said, &quot;The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous man.&quot;
**''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 54.

== Ibn Rajab al-Hanbali ==

* '''Allah loves that His servants ask Him for all their religious and worldly needs, like their food, drink, clothing, etc.,''' just as they ask Him for His guidance and mercy, because if the servant asks for everything he needs from Allah, he has shown his need of Him, and Allah loves that.
** Jami al-Ulum wa al-Hikam, vol. 2, p. 38.

== Ibn Muflih al-Hanbali ==

* How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.'''
** Al-Adab al-Shar'iyyah (3/240)

== Muhammad Taqi al-Din al-Hilali ==

* The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: &quot;And We would never punish until We had sent a Messenger&quot; [17:15].
** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Abdul-Rahman al-Sa'di ==

* This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it.
** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Muhammad Nasir al-Din al-Albani ==

* What matters is not doing a lot of worship. What matters is that the worship follows the Sunnah and stays far from innovation. Ibn Mas'ud, may Allah be pleased with him, pointed to this when he said: &quot;Doing a little while following the Sunnah is better than doing a lot while following an innovation.&quot;
** Al-Silsilah al-Sahihah (5/14), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibn Taymiyya ==

=== Refuting the Murji'ah ===
* Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills.
** As-Sarim al-Maslul, 2/339

== Abu Bakr al-Marrudhi ==

* I heard Abu Abdillah Ahmad ibn Hanbal, may Allah have mercy upon him, say: &quot;[My wife] Umm Salih was with me for 30 years, and we never disagreed in a single word.&quot;
** Tarikh Baghdad (14/438), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Sufyan al-Thawri ==

* I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity.
** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey]

== Ibrahim al-Taymi ==

* Two things have cut me off from the pleasure of this world: remembering death, and remembering standing before Allah the Exalted.
** Al-Tadhkirah, p. 125, [Al-Baseerah Islamic Center in Plainfield, New Jersey]

[[Category:Islamic Studies]]
[[Category:Atcovi's Work]]</text>
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    <title>Liouville theory with a central charge less than one</title>
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[[w:Liouville theory]] is a simple 2d conformal field theory, with applications to quantum gravity, string theory, and statistical physics. The theory can be solved using either a constructive, path integral approach, or the axiomatic conformal bootstrap approach. The path integral approach can be made mathematically rigorous in the context of probability theory, at the price of restricting the central charge to &lt;math display=&quot;inline&quot;&gt;c\geq 25&lt;/math&gt;. The bootstrap approach provides a unique solution in the larger domain &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, and numerically tests its consistency.

Correlation functions of Liouville theory are meromorphic functions of &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty , 1)&lt;/math&gt;, with an essential singularity for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. This singularity occurs when the degenerate values &lt;math display=&quot;inline&quot;&gt;\{P_{(r,s)}=\frac12(r\beta -s\beta^{-1})\}_{r,s\in\mathbb{N}^*}&lt;/math&gt; of the momentum become dense on the real line, instead of being isolated from one another. (Notations for the central charge &lt;math display=&quot;inline&quot;&gt;c=13-6\beta^2-6\beta^{-2}&lt;/math&gt; and conformal dimension &lt;math display=&quot;inline&quot;&gt;\Delta=\frac{c-1}{24}+P^2&lt;/math&gt;.) Due to this singularity, no theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; can be deduced from the original definition of Liouville theory.

However, in the conformal bootstrap approach, the assumptions and methods that lead to Liouville theory can be applied for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;. The result is a unique conformal field theory with a continuous spectrum &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{R}&lt;/math&gt;.&lt;ref name=&quot;rs15&quot;/&gt; This theory has been called &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory, or imaginary Liouville theory, or timelike Liouville theory. (The latter name originated before the spectrum was found to be spacelike.)

In this page we review the conformal bootstrap solution, and discuss whether its features can emerge from a path integral construction. In particular, we argue that the behaviour of fields when their dimension goes to zero is hard to reproduce from the path integral.  

== Features of correlation functions ==

=== Spectrum and 3-point structure constant ===

In the conformal bootstrap approach, solving Liouville theory reduces to determining the spectrum and 3-point structure constant.

The '''spectrum''' is the set of values of the momentum over which we integrate (or sum) when computing operator product expansions. The spectrum of Liouville theory is continuous for any &lt;math display=&quot;inline&quot;&gt;c\in\mathbb{C}&lt;/math&gt;, and OPEs read &lt;math display=&quot;inline&quot;&gt;V_{P_1}V_{P_2}\sim \int_\mathbb{R} dP\ V_P&lt;/math&gt;. We can compute correlation functions of fields that do not belong to the spectrum: fields &lt;math display=&quot;inline&quot;&gt;V_P&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, and degenerate fields &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;r,s\in\mathbb{N}^*&lt;/math&gt;. The field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; has momentum &lt;math display=&quot;inline&quot;&gt;P_{(r,s)}&lt;/math&gt;, and the OPE &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}V_P&lt;/math&gt; involves a finite number &lt;math display=&quot;inline&quot;&gt;rs&lt;/math&gt; of primary fields. In particular, the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle 1,1\rangle}&lt;/math&gt; with momentum &lt;math display=&quot;inline&quot;&gt;P=P_{(1,1)}&lt;/math&gt; is the identity field, and obeys the singular vector equation &lt;math display=&quot;inline&quot;&gt;\frac{\partial}{\partial z} V^d_{\langle 1,1\rangle}(z)=0&lt;/math&gt;.

The '''3-point structure constant''' is a function of 3 momentums &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt;. To constrain it, we study 4-point correlation functions with degenerate fields &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 2,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,2\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle&lt;/math&gt;. Crossing symmetry and single-valuedness of these 4-point functions imply shift equations, which determine how &lt;math display=&quot;inline&quot;&gt;C_{P_1,P_2,P_3}&lt;/math&gt; behaves under &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P_k\to P_k+\beta^{-1}&lt;/math&gt;.

Shift equations have a unique solution that is meromorphic in &lt;math display=&quot;inline&quot;&gt;\beta&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;, called the DOZZ formula for Dorn–Otto and Zamolodchikov–Zamolodchikov. There is also a unique solution that is meromorphic over &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(25, \infty)&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;P\in\mathbb{C}&lt;/math&gt;. The latter solution is relevant to &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.

However, the 3-point structure constant does not known the spectrum of the theory. It also does not know its domain of definition in the complex &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;-plane. In fact it does not even know which theory we are studying: the 3-point structure constant of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory also appears in critical loop models, which have a discrete spectrum and are defined over the half-plane &lt;math display=&quot;inline&quot;&gt;\Re c&lt;13&lt;/math&gt;.&lt;ref name=&quot;rib24&quot;/&gt;

=== Features of 4-point functions ===

To understand Liouville theory with &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, it is crucial to study 4-point functions, whose decompositions into conformal blocks are integrals over the spectrum. This determines the spectrum, and explains why the two definitions of Liouville theory for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt; are not related by analytic continuation. And this leads to some features of a 4-point function &lt;math display=&quot;inline&quot;&gt;\left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle&lt;/math&gt;, which any proposed construction would have to reproduce:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' Due to the continuous spectrum, the leading behaviour involves a logarithmic factor: &lt;math display=&quot;block&quot;&gt;\begin{align}
  \left\langle \prod_{i=1}^4V_{P_i}(z_i)\right\rangle \underset{z_1\to z_2}{\sim} |z_1-z_2|^{-2P_1^2-2P_2^2-\frac{c-1}{12}} \left|\log|z_1-z_2|\right|^{\alpha}\ ,
\end{align}&lt;/math&gt; where &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac12&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and &lt;math display=&quot;inline&quot;&gt;\alpha=-\frac32&lt;/math&gt; otherwise.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' The 4-point function is a holomorphic function of &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt; for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, and meromorphic otherwise. This is because the structure constants and conformal blocks are themselves holomorphic or meromorphic.&lt;/p&gt;
&lt;p&gt;These properties hold under the assumption that fields are well-normalized. In the bootstrap approach, we are indeed free to renormalize fields as &lt;math display=&quot;inline&quot;&gt;V_P(z)\to \lambda_PV_P(z)&lt;/math&gt; for any function &lt;math display=&quot;inline&quot;&gt;\lambda_P&lt;/math&gt;, which might have singularities. In correlation functions, we will call factorized the singularities that can be added or removed by field renormalizations.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' The limit &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P&lt;/math&gt; is the degenerate field &lt;math display=&quot;inline&quot;&gt;V^d_{\langle r,s\rangle}&lt;/math&gt; unless &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, in which case &lt;math display=&quot;inline&quot;&gt;\lim_{P\to P_{(r,s)}}V_P=V_{P_{(r,s)}}&lt;/math&gt;. (Same story with &lt;math display=&quot;inline&quot;&gt;P\to -P_{(r,s)}&lt;/math&gt;.) In particular, for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, any &lt;math display=&quot;inline&quot;&gt;N\geq 3&lt;/math&gt;-point function &lt;math display=&quot;inline&quot;&gt;\left\langle V_{P_{(1,1)}}(z)\cdots \right\rangle&lt;/math&gt; depends on &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;, in contrast to &lt;math display=&quot;inline&quot;&gt;\left\langle V^d_{\langle 1,1\rangle}(z)\cdots \right\rangle&lt;/math&gt; which does not.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

These features follow from the known spectrum and structure constant, together with the convergence of the decomposition of correlation functions into structure constants and conformal blocks. This decomposition is an integral over &lt;math display=&quot;inline&quot;&gt;P&lt;/math&gt; that is numerically seen to converge very fast. Failure to converge would call into question the very existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory. Summary table: &lt;math display=&quot;block&quot;&gt;\begin{align}
 \begin{array}{|c|r||c|c|}
  \hline
  \# &amp; \text{Property} &amp; c\in (-\infty, 1) &amp; c\in\mathbb{C}\backslash (-\infty, 1)
  \\
  \hline
  1 &amp; \text{Log exponent for }z_1\to z_2 &amp; -\frac12 &amp; -\frac32
  \\
  \hline
  2 &amp; \text{Analyticity in } P_i &amp; \text{holomorphic} &amp; \text{meromorphic}
  \\
  \hline
  3&amp; \lim_{P\to P_{(r,s)}}V_P &amp; V_{P_{(r,s)}} &amp; V^d_{\langle r,s\rangle}
  \\
  \hline
 \end{array}
\end{align}&lt;/math&gt;

== Path integral construction ==

=== Definition ===

In known constructions of Liouville theory, &lt;math display=&quot;inline&quot;&gt;N&lt;/math&gt;-point functions are built as a path integral of the type &lt;math display=&quot;block&quot;&gt;\begin{align}
 \left\langle \prod_{i=1}^N V_{P_i}(z_i) \right\rangle=\int D\varphi\ \prod_{i=1}^N e^{2(P_i+P_{(1,1)})\varphi(z_i)}\ ,
\end{align}&lt;/math&gt; for some field (or random variable) &lt;math display=&quot;inline&quot;&gt;\varphi(z)&lt;/math&gt;, with a functional measure &lt;math display=&quot;inline&quot;&gt;D\varphi&lt;/math&gt; that includes the action of the theory. A given contruction will work under some conditions on the central charge &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; and the momentums &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, such as:

* The restrictions &lt;math display=&quot;inline&quot;&gt;c\in (25,\infty)&lt;/math&gt; or &lt;math display=&quot;inline&quot;&gt;c\in (-\infty, 1)&lt;/math&gt;: positivity conditions that allows probabilistic methods to apply.
* Seiberg bounds on momentums, for convergence of the integral over &lt;math display=&quot;inline&quot;&gt;\varphi&lt;/math&gt;.
* Another condition on momentums called charge neutrality, for constructions based on free fields.

A simple analogy is the construction of the function &lt;math display=&quot;inline&quot;&gt;f(z)=\frac{1}{1-z}&lt;/math&gt; as the infinite series &lt;math display=&quot;inline&quot;&gt;\sum_{n=0}^\infty z^n&lt;/math&gt;, where the pole of &lt;math display=&quot;inline&quot;&gt;f(z)&lt;/math&gt; at &lt;math display=&quot;inline&quot;&gt;z=1&lt;/math&gt; manifests itself as the condition &lt;math display=&quot;inline&quot;&gt;|z|&lt;1&lt;/math&gt; on the construction. Similarly, we expect that the conditions for constructions of Liouville theory are related to the singularities of correlation functions.

=== Reproducing known features ===

Existing attempts at constructing &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory use similar ideas as the construction that works for &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Features of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory are therefore harder to reproduce when they differ more from features at &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;. Let us discuss the features we discussed previously:

&lt;ol&gt;
&lt;li&gt;&lt;p&gt;'''Limit of coinciding fields:''' This feature does not depend much on the value of &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;, and does not look very problematic. All the same, it would be nice to check it.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Analyticity in momentums:''' In the case of the 3-point function for &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, the known constructions do not yield the well-normalized, holomorphic formula: there are factorized poles at &lt;math display=&quot;inline&quot;&gt;2P_i\in (-\beta\mathbb{N}^*-\beta^{-1}\mathbb{N})\cup (\beta\mathbb{N}+\beta^{-1}\mathbb{N}^*)&lt;/math&gt;.&lt;/p&gt;
&lt;p&gt;Since the integrand of the path integral is holomorphic in &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;, we expect that the integral is locally holomorphic, wherever it converges. We expect that the domain of convergence is related to the factorized poles, and should therefore include all (or part) of the interval &lt;math display=&quot;inline&quot;&gt;P_i\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt;.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;'''Degenerate values of momentums:''' In a path integral construction, if the limit commutes with the integral &lt;math display=&quot;inline&quot;&gt;\int D\varphi&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;\lim_{P\to -P_{(1,1)}} V_{P}(z)=V^d_{\langle 1,1\rangle}(z)&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independent. For &lt;math display=&quot;inline&quot;&gt;c\in \mathbb{C}\backslash(-\infty, 1)&lt;/math&gt;, this &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-independence is indeed observed. (In particular, the leading behaviour at &lt;math display=&quot;inline&quot;&gt;z_1\to z_2&lt;/math&gt; comes with a &lt;math display=&quot;inline&quot;&gt;P_i&lt;/math&gt;-dependent prefactor that goes to zero if &lt;math display=&quot;inline&quot;&gt;P_i\to -P_{(1,1)}&lt;/math&gt;.) For &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt;, since &lt;math display=&quot;inline&quot;&gt;-P_{(1,1)}\in (-\frac{\beta}{2},\frac{\beta^{-1}}{2})&lt;/math&gt; for any &lt;math display=&quot;inline&quot;&gt;\beta\in\mathbb{R}_{&gt;0}&lt;/math&gt;, there is no reason for the construction to fail for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. Nevertheless, fail it must, since the &lt;math display=&quot;inline&quot;&gt;z&lt;/math&gt;-dependence survives in the limit.&lt;/p&gt;&lt;/li&gt;&lt;/ol&gt;

Conclusion:

&lt;blockquote&gt;Any path integral construction is likely valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;. And if it is valid for &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt;, then it cannot describe &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory.
&lt;/blockquote&gt;

=== Specific attempt === 

An attempt to construct &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is due to Usciati, Guillarmou, Rhodes and Santachiara.&lt;ref name=&quot;urgs25&quot;/&gt; In that article, the focus was on the 3-point function, in which case &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is outside the domain of validity. The same approach was extended to &lt;math display=&quot;inline&quot;&gt;N\geq 4&lt;/math&gt;-point functions by Cerclé and Usciati,&lt;ref name=&quot;cu26&quot;/&gt; and it seems that &lt;math display=&quot;inline&quot;&gt;P=-P_{(1,1)}&lt;/math&gt; is allowed. This would imply that this construction does not agree with the known theory. The authors reach the same conclusion by comparing spectrums, but stop short of claiming that they have found a new CFT.

== Living without a construction == 

Even for CFTs whose correlation functions can be computed in the bootstrap approach, a construction can be interesting for two main reasons:

* Constructing a CFT proves its existence, which is hard to do in the bootstrap approach.
* The Lagrangian is a simple object that encodes broad features of the theory, such as its relations to other theories. This can be very useful in applications.

What can be done in &lt;math&gt;c\leq 1&lt;/math&gt; Liouville theory, in the absence of a construction?

=== Proof of existence ===

Proving the existence of the theory amounts to proving crossing symmetry of all correlation functions. Since we have explicit expressions in terms of structure constants and conformal blocks, this is a well-defined technical problem. A construction could provide an elegant solution, but other approaches are possible. In particular, from the consistency of diagonal minimal models, it might be possible to deduce the existence of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory by taking limits.

=== Application to the Virasoro minimal string ===

A recent application of &lt;math display=&quot;inline&quot;&gt;c\leq 1&lt;/math&gt; Liouville theory is the construction of the Virasoro minimal string.&lt;ref name=&quot;cemr23&quot;/&gt; This application did not need more than the bootstrap definition of the theory, including for finding a matrix integral description of the resulting string theory. A construction would have been convenient, but it was possible to do without it. 

When it came to deducing correlation numbers of the string theory from correlation functions of the CFT, it was essential to know their analytic properties extensively:&lt;ref name=&quot;kl25&quot;/&gt; a construction that would come with limitations on the allowed values of momentums might not have been very helpful.

== References ==

{{Reflist|refs=
&lt;ref name=&quot;kl25&quot;&gt;{{cite web | last1=Khromov | first1=Dmitry | last2=Litvinov | first2=Alexey | title=On correlation numbers $V_{0,4}$ and $V_{1,1}$ in Virasoro Minimal String Theory | website=arXiv.org | date=2025-09-30 | url=https://arxiv.org/abs/2509.25960 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;cemr23&quot;&gt;{{cite journal | last1=Collier | first1=Scott | last2=Eberhardt | first2=Lorenz | last3=Mühlmann | first3=Beatrix | last4=Rodriguez | first4=Victor A. | title=The Virasoro minimal string | journal=SciPost Physics | publisher=Stichting SciPost | volume=16 | issue=2 | date=2024-02-26 | issn=2542-4653 | doi=10.21468/scipostphys.16.2.057 | doi-access=free | url=https://arxiv.org/abs/2309.10846}}&lt;/ref&gt;
&lt;ref name=&quot;cu26&quot;&gt;{{cite web | last1=Cerclé | first1=Baptiste | last2=Usciati | first2=Romain | title=On the local conformal structure of Imaginary Liouville theory | website=arXiv.org | date=2026-08-03 | url=https://arxiv.org/abs/2608.02543 | access-date=2026-11-06}}&lt;/ref&gt;
&lt;ref name=&quot;urgs25&quot;&gt;{{cite journal | last1=Usciati | first1=Romain | last2=Guillarmou | first2=Colin | last3=Rhodes | first3=Remi | last4=Santachiara | first4=Raoul | title=Probabilistic Construction of Noncompactified Imaginary Liouville Field Theory | journal=Physical Review Letters | volume=136 | issue=3 | date=2026-01-21 | issn=0031-9007 | doi=10.1103/d723-gk4m | doi-access=free | url=https://arxiv.org/abs/2505.09390}}&lt;/ref&gt;
&lt;ref name=&quot;rib24&quot;&gt;{{cite journal | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | url = https://arxiv.org/abs/2411.17262| date=2026-07-22 | issn=2590-1990 | doi=10.21468/scipostphyslectnotes.130 | doi-access=free | page=}}&lt;/ref&gt;
&lt;ref name=&quot;rs15&quot;&gt;{{cite journal | last1=Ribault | first1=Sylvain | last2=Santachiara | first2=Raoul | title=Liouville theory with a central charge less than one | journal=Journal of High Energy Physics | volume=2015 | issue=8 | year=2015 | issn=1029-8479 | doi=10.1007/jhep08(2015)109 | doi-access=free | url = https://arxiv.org/abs/1503.02067| access-date=2026-11-06 | page=}}&lt;/ref&gt;
}}


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A scorecard is the written record of a cricket innings. Each batter's runs, balls faced and manner of dismissal are set out in turn, and a team's total is written as runs for the wickets lost, so that a total shown as 112/3 is read as 112 runs for three wickets.

Extras such as wides and no-balls are added to the batting side's total and recorded separately from runs scored off the bat.

==See also==
* [[Cricket]]

[[Category:Cricket]]</text>
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== 2026-11-09 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&amp;#126;2026-59650-28|your recent contributions]] have been undone because you removed content without adequately explaining why. In the future, it would be helpful to others if you described your changes to &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; with an accurate [[:m:en:Help:Edit summary|edit summary]]. If this was a mistake, don't worry; the removed content has been restored. If you would like to experiment, please use the sandbox. Thanks. &lt;/div&gt;&lt;!-- Glow-delete1 @ 1794228212165s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:43, 9 November 2026 (UTC)</text>
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    <title>User talk:~2026-59426-49</title>
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      <timestamp>2026-11-09T12:46:53Z</timestamp>
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== 2026-11-09 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&amp;#126;2026-59426-49|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1794228410277.7s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:46, 9 November 2026 (UTC)</text>
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    <title>File:VLSI.Arith.2B.CLA.20261109.pdf</title>
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|Description=Carry Lookahead Adders 2B traditional (20261109 - 20261107)
|Source={{own|Young1lim}}
|Date=2026-11-09
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{{Information
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|Source={{own|Young1lim}}
|Date=2026-11-09
|Author=Young W. Lim
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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||Description=Block CLA 2C Multi Level  (20261109 - 20261107)
|Source={{own|Young1lim}}
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{{Information
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|Source={{own|Young1lim}}
|Date=2026-11-09
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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    <title>User talk:~2026-59632-72</title>
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== 2026-11-09 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&amp;#126;2026-59632-72|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1794228744042.1s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:52, 9 November 2026 (UTC)</text>
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== 2026-11-09 ==
&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; style=&quot;text-align: left&quot; lang=&quot;en&quot;&gt;[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/&amp;#126;2026-59286-66|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt; as a &quot;soapbox&quot;]] are not permitted. Take a look at the welcome pages to learn more about &lt;span style=&quot;white-space:nowrap&quot;&gt;Wikiversity&lt;/span&gt;. Thanks. &lt;/div&gt;&lt;!-- Glow-advert1 @ 1794229064053.8s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:57, 9 November 2026 (UTC)</text>
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      <comment>{{Information
|Description=C04.SA0: Address and Dereference Operators (20261109 - 20261107)
|Source={{own|Young1lim}}
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|Author=Young W. Lim
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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== 2026-11-09 ==
[[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you.&lt;!-- Glow-vandalism1 @ 1794230593381.7s --&gt;&lt;nowiki&gt;&lt;/nowiki&gt; [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:23, 9 November 2026 (UTC)</text>
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